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Approach to a Proof of the Riemann Hypothesis
by the Second Mean-Value Theorem of Calculus
Alfred W¨unsche
formerly: Humboldt-Universit¨at Berlin,
Institut f¨ur Physik, Nichtklassische Strahlung (MPG),
Newtonstrasse 15, 12489 Berlin, Germany
e-mail: alfred.wuensche@physik.hu-berlin.de
Abstract
By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class
of functions Ξ(z) with an integral representation of the form R+∞
0du Ω(u) ch(uz) with a real-
valued function Ω(u)≥0 which is non-increasing and decreases in infinity more rapidly than
any exponential functions exp (−λu), λ > 0 possesses zeros only on the imaginary axis. The
Riemann zeta function ζ(s) as it is known can be related to an entire function ξ(s) with the same
non-trivial zeros as ζ(s). Then after a trivial argument displacement s↔z=s−1
2we relate it
to a function Ξ(z) with a representation of the form Ξ(z) = R+∞
0du Ω(u) ch(uz) where Ω(u) is
rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the
position of its zeros on the imaginary axis z= iyby the second mean-value theorem. Besides
this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form
derived in the Appendix B. All this means that we prove a theorem for zeros of Ξ(z) on the
imaginary axis z= iyfor a whole class of function Ω(u) which includes in this way the proof of
the Riemann hypothesis. This whole class includes, in particular, the modified Bessel functions
Iν(z) for which it is known that their zeros lie on the imaginary axis and which affirms our
conclusions that we intend to publish at another place. In the same way a class of almost-
periodic functions to piece-wise constant nonincreasing functions Ω(u) belong also to this case.
At the end we give shortly an equivalent way of a more formal description of the obtained results
using the Mellin transform of functions with its variable substituted by an operator.
1. Introduction
The Riemann zeta function ζ(s) which basically was known already to Euler establishes the most important
link between number theory and analysis. The proof of the Riemann hypothesis is a longstanding problem
since it was formulated by Riemann [1] in 1859. The Riemann hypothesis is the conjecture that all nontrivial
zeros of the Riemann zeta function ζ(s) for complex s=σ+itare positioned on the line s=1
2+itthat means
on the line parallel to the imaginary axis through real value σ=1
2in the complex plane and in extension
that all zeros are simple zeros [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] (with extensive lists of
references in some of the cited sources, e.g., [4, 5, 9, 12, 14]). The book of Edwards [5] is one of the best older
sources concerning most problems connected with the Riemann zeta function. There are also mathematical
tables and chapters in works about Special functions which contain information about the Riemann zeta
function and about number analysis, e.g., Whittaker and Watson [2] (chap. 13), Bateman and Erd´elyi [18]
(chap. 1) about zeta functions, [19] (chap. 17) about number analysis, and Apostol [20, 21] (chaps. 25 and
27). The book of Borwein, Choi, Rooney and Weirathmueller [12] gives on the first 90 pages a short account
about achievements concerning the Riemann hypothesis and its consequences for number theory and on the
following about 400 pages it reprints important original papers and expert witnesses in the field. Riemann
has put aside the search for a proof of his hypothesis ’after some fleeting vain attempts’ and emphasizes that
1
arXiv:1607.04551v1 [math.GM] 9 Jul 2016
’it is not necessary for the immediate objections of his investigations’ [1] (see [5]). The Riemann hypothesis
was taken by Hilbert as the 8-th problem in his representation of 23 fundamental unsolved problems in
pure mathematics and axiomatic physics in a lecture hold on 8 August in 1900 at the Second Congress of
Mathematicians in Paris [22, 23]. The vast experience with the Riemann zeta function in the past and the
progress in numerical calculations of the zeros (see, e.g., [5, 10, 11, 16, 17, 24, 25]) which all confirmed the
Riemann hypothesis suggest that it should be true corresponding to the opinion of most of the specialists in
this field but not of all specialists (arguments for doubt are discussed in [26]).
The Riemann hypothesis is very important for prime number theory and a number of consequences is
derived under the unproven assumption that it is true. As already said a main role plays a function ζ(s)
which was known already to Euler for real variables sin its product representation (Euler product) and
in its series representation and was continued to the whole complex s-plane by Riemann and is now called
Riemann zeta function. The Riemann hypothesis as said is the conjecture that all nontrivial zeros of the
zeta function ζ(s) lie on the axis parallel to the imaginary axis and intersecting the real axis at s=1
2. For
the true hypothesis the representation of the Riemann zeta function after exclusion of its only singularity at
s= 1 and of the trivial zeros at s=−2n, (n= 1,2, . . .) on the negative real axis is possible by a Weierstrass
product with factors which only vanish on the critical line σ=1
2. The function which is best suited for this
purpose is the so-called xi function ξ(s) which is closely related to the zeta function ζ(s) and which was
also introduced by Riemann [1]. It contains all information about the nontrivial zeros and only the exact
positions of the zeros on this line are not yet given then by a closed formula which, likely, is hardly to find
explicitly but an approximation for its density was conjectured already by Riemann [1] and proved by von
Mangoldt [27]. The ”(pseudo)-random” character of this distribution of zeros on the critical line remembers
somehow the ”(pseudo)-random” character of the distribution of primes where one of the differences is that
the distribution of primes within the natural numbers becomes less dense with increasing integers whereas
the distributions of zeros of the zeta function on the critical line becomes more dense with higher absolute
values with slow increase and approaches to a logarithmic function in infinity.
There are new ideas for analogies to and application of the Riemann zeta function in other regions of
mathematics and physics. One direction is the theory of random matrices [16, 24] which shows analogies in
their eigenvalues to the distribution of the nontrivial zeros of the Riemann zeta function. Another interesting
idea founded by Voronin [28] (see also [16, 29]) is the universality of this function in the sense that each
holomorphic function without zeros and poles in a certain circle with radius less 1
2can be approximated
with arbitrary required accurateness in a small domains of the zeta function to the right of the critical line
within 1
2≤s≤1. An interesting idea is elaborated in articles of Neuberger, Feiler, Maier and Schleich
[30, 31]. They consider a simple first-order ordinary differential equation with a real variable t(say the time)
for given arbitrary analytic functions f(z) where the time evolution of the function for every point zfinally
transforms the function in one of the zeros f(z) = 0 of this function in the complex z-plane and illustrate
this process graphically by flow curves which they call Newton flow and which show in addition to the zeros
the separatrices of the regions of attraction to the zeros. Among many other functions they apply this to
the Riemann zeta function ζ(z) in different domains of the complex plane. Whether, however, this may lead
also to a proof of the Riemann hypothesis is more than questionable.
Number analysis defines some functions of a continuous variable, for example, the number of primes
π(x) less a given real number xwhich last is connected with the discrete prime number distribution (e.g.,
[3, 4, 5, 7, 9, 11]) and establishes the connection to the Riemann zeta function ζ(s). Apart from the product
representation of the Riemann zeta function the representation by a type of series which is now called
Dirichlet series was already known to Euler. With these Dirichlet series in number theory are connected
some discrete functions over the positive integers n= 1,2, . . . which play a role as coefficients in these series
and are called arithmetic functions (see, e.g., Chandrasekharan [4] and Apostol [13]). Such functions are the
M¨obius function µ(n) and the Mangoldt function Λ(n) as the best known ones. A short representation of
the connection of the Riemann zeta function to number analysis and of some of the functions defined there
became now standard in many monographs about complex analysis (e.g., [15]).
Our means for the proof of the Riemann hypothesis in present article are more conventional and ”old-
fashioned” ones, i.e. the Real Analysis and the Theory of Complex Functions which were developed already
for a long time. The most promising way for a proof of the Riemann hypothesis as it seemed to us in past
2
is via the already mentioned entire function ξ(s) which is closely related to the Riemann zeta function ζ(s).
It contains all important elements and information of the last but excludes its trivial zeros and its only
singularity and, moreover, possesses remarkable symmetries which facilitate the work with it compared with
the Riemann zeta function. This function ξ(s) was already introduced by Riemann [1] and dealt with, for
example, in the classical books of Titchmarsh [3], Edwards [5] and in almost all of the sources cited at the
beginning. Present article is mainly concerned with this xi function ξ(s) and its investigation in which, for
convenience, we displace the imaginary axis by 1
2to the right that means to the critical line and call this Xi
function Ξ(z) with z=x+ iy. We derive some representations for it among them novel ones and discuss its
properties, including its derivatives, its specialization to the critical line and some other features. We make
an approach to this function via the second mean value theorem of analysis (Gauss-Bonnet theorem, e.g.,
[37, 38]) and then we apply an operator identity for analytic functions which is derived in Appendix B and
which is equivalent to a somehow integrated form of the Cauchy-Riemann equations. This among other not
so successful trials (e.g., via moments of function Ω(u)) led us finally to a proof of the Riemann hypothesis
embedded into a proof for a more general class of functions.
Our approach to a proof of the Riemann hypothesis in article in rough steps is as follows:
First we shortly represent the transition from the Riemann zeta function ζ(s) of complex variable s=σ+ it
to the xi function ξ(s) introduced already by Riemann and derive for it by means of the Poisson summation
formula a representation which is convergent in the whole complex plane (Section 2 with main formal part
in Appendix A). Then we displace the imaginary axis of variable sto the critical line at s=1
2+ itby
s→z=s−1
2that is purely for convenience of further working with the formulae. However, this has also
the desired subsidiary effect that it brings us into the fairway of the complex analysis usually represented
with the complex variable z=x+ iy. The transformed ξ(s) function is called Ξ(z) function.
The function Ξ(z) is represented as an integral transform of a real-valued function Ω(u) of the real
variable uin the form Ξ(z) = R+∞
0du Ω(u) ch(uz) which is related to a Fourier transform (more exactly to
Cosine Fourier transform). If the Riemann hypothesis is true then we have to prove that all zeros of the
function Ξ(z) occur for x= 0.
To the Xi function in mentioned integral transform we apply the second mean-value theorem of real
analysis first on the imaginary axes and discuss then its extension from the imaginary axis to the whole
complex plane. For this purpose we derive in Appendix B in operator form relations of the complex mean-
value parameter in our application of the second mean-value theorem to this parameter on the imaginary
axis which are equivalents in integral form to the Cauchy-Riemann equations in differential form and apply
this in specific form to the Xi function (Sections 3 and 4).
Then in Section 5 we accomplish the proof with the discussion and solution of the two most important
equations (5.10) and (5.11) for the last and decisive stage of the proof. These two equations are derived
before in preparation of this last stage of the proof. From these equations it is seen that the obtained two real
equations admit zeros of the Xi function only on the imaginary axis. This proves the Riemann hypothesis
by the equivalence of the Riemann zeta function ζ(s) to the Xi function Ξ(z) and embeds it into a whole
class of functions with similar properties and positions of their zeros.
The Sections 6-7 serve for illustrations and graphical representations of the specific parameters (e.g.,
mean-value parameters) for the Xi function to the Riemann hypothesis and for other functions which in our
proof by the second mean-value problem are included for the existence of zeros only on the imaginary axis.
This is in particular the whole class of modified Bessel functions Iν(z),−1
2< ν < +∞with real indices ν
which possess zeros only on the imaginary axis yand where a proof by means of the differential equations
exists.
2. From Riemann zeta function ζ(s) to related xi function ξ(s) and
its argument displacement to function Ξ(z)
In this Section we represent the known transition from the Riemann zeta function ζ(s) to a function ξ(s)
and finally to a function Ξ(z) with displaced complex variable s→z=s−1
2for rational effective work
3
and establish some of the basic representations of these functions, in particular, a kind of modified Cosine
Fourier transformations of a function Ω(u) to the function Ξ(z).
As already expressed in the Introduction, the most promising way for a proof of the Riemann hypothesis
as it seems to us is the way via a certain integral representation of the related xi function ξ(s). We sketch here
the transition from the Riemann zeta function ζ(s) to the related xi function ξ(s) in a short way because,
in principle, it is known and we delegate some aspects of the derivations to Appendix A
Usually, the starting point for the introduction of the Riemann zeta function ζ(s) is the following relation
between the Euler product and an infinite series continued to the whole complex s-plane
ζ(s)≡
∞
Y
n=1 1−1
ps
n−1
=
∞
X
n=1
1
ns,(σ≡Re(s)>1) ,(2.1)
where pndenotes the ordered sequence of primes (p1= 2, p2= 3, p3= 5, . . .). The transition from the
product formula to the sum representation in (2.1) via transition to the Logarithm of ζ(s) and Taylor
series expansion of the factors log 1−1
ps
n−1in powers of 1
ps
nusing the uniqueness of the prime-number
decomposition is well known and due to Euler in 1737. It leads to a special case of a kind of series later
introduced and investigated in more general form and called Dirichlet series. The Riemann zeta function
ζ(s) can be analytically continued into the whole complex plane to a meromorphic function that was made
and used by Riemann. The sum in (2.1) converges uniformly for complex variable s=σ+ itin the open
semi-planes with arbitrary σ > 1 and arbitrary t. The only singularity of the function ζ(s) is a simple pole
at s= 1 with residue 1 that we discuss below.
The product form (2.1) of the zeta function ζ(s) shows that it involves all prime numbers pnexactly one
times and therefore it contains information about them in a coded form. It proves to be possible to regain
information about the prime number distribution from this function. For many purposes it is easier to work
with meromorphic and, moreover, entire functions than with infinite sequences of numbers but in first case
one has to know the properties of these functions which are determined by their zeros and their singularities
together with their multiplicity.
From the well-known integral representation of the Gamma function
Γ(z) = Z+∞
0
dt tz−1e−t,(Re(z)>0) ,(2.2)
follows by the substitutions t=nµx, µz =swith an appropriately fixed parameter µ > 0 for arbitrary
natural numbers n
1
ns=1
Γs
µZ+∞
0
dx x s
µ−1e−nµx,Re s
µ>0.(2.3)
Inserting this into the sum representation (2.1) and changing the order of summation and integration, we
obtain for choice µ= 1 of the parameter using the sum evaluation of the geometric series
ζ(s) = 1
Γ(s)Z+∞
0
dp ps−1
ep−1,(Re(s)>1) ,(2.4)
and for choice µ= 2 with substitution p=πq2of the integration variable (see [1] and, e.g., [3, 4, 5, 7, 9])
ζ(s) = πs
2s
Γs
2+ 1Z+∞
0
dq qs−1
∞
X
n=1
exp −πn2q2,(Re(s)>1) .(2.5)
Other choice of µseems to be of lesser importance. Both representations (2.4) and (2.5) are closely related
to a Mellin transform ˆ
f(s) of a function f(t) which together with its inversion is generally defined by (e.g.,
4
[15, 32, 33, 34, 35])
f(t)→ˆ
f(s)≡Z+∞
0
dt ts−1f(t),⇒f(λt)→1
λsˆ
f(s),(λ > 0 ) ,
ˆ
f(s)→f(t) = 1
i2πZc+i∞
c−i∞
ds t−sˆ
f(s),⇒ˆ
f(s−s0)→ts0f(t),(2.6)
where cis an arbitrary real value within the convergence strip of ˆ
f(s) in complex s-plane. The Mellin trans-
form ˆ
f(s) of a function f(t) is closely related to the Fourier transform ˜ϕ(y) of the function ϕ(x)≡f(ex) by
variable substitution t= exand y= is. Thus the Riemann zeta function ζ(s) can be represented, substan-
tially (i.e., up to factors depending on s), as the Mellin transforms of the functions f(t) = P∞
n=1 e−nt =1
et−1
or of f(t) = P∞
n=1 exp −πn2t2, respectively. The kernels of the Mellin transform are the eigenfunctions of
the differential operator t∂
∂t to eigenvalue s−1 or, correspondingly, of the integral operator exp α t ∂
∂t of
the multiplication of the argument of a function by a factor eα(scaling of argument). Both representations
(2.4) and (2.5) can be used for the derivation of further representations of the Riemann zeta function and
for the analytic continuation. The analytic continuation of the Riemann zeta function can also be obtained
using the Euler-Maclaurin summation formula for the series in (2.1) (e.g., [5, 11, 15]).
Using the Poisson summation formula, one can transform the representation (2.5) of the Riemann zeta
function to the following form
ζ(s) = πs
2
s
2!(s−1) (1
2−s(1 −s)Z+∞
1
dq qs+q1−s
q
∞
X
n=1
exp −πn2q2).(2.7)
This is known [1, 3, 4, 5, 7, 9] but for convenience and due to the importance of this representation for our
purpose we give a derivation in Appendix A. From (2.7) which is now already true for arbitrary complex
sand, therefore, is an analytic continuation of the representations (2.1) or (2.5) we see that the Riemann
zeta function satisfies a functional equation for the transformation of the argument s→1−s. In simplest
form it appears by ’renormalizing’ this function via introduction of the xi function ξ(s) defined by Riemann
according to [1] and to [5, 20]1
ξ(s)≡(s−1) s
2!
πs
2
ζ(s),(2.8)
and we obtain for it the following representation converging in the whole complex plane of s(e.g., [1, 4, 5, 9, 7])
ξ(s) = 1
2−s(1 −s)Z+∞
1
dp ps+p1−s
p
∞
X
n=1
exp −πn2p2,(2.9)
with the ’normalization’
ξ(0) = ξ(1) = −ζ(0) = 1
2.(2.10)
For s=1
2the xi function and the zeta function possess the (likely transcendental) values
ξ1
2=−1
4!
2π1
4
ζ1
2= 0.4971207782, ζ 1
2=−1.4603545088 (2.11)
Contrary to the Riemann zeta function ζ(s) the function ξ(s) is an entire function. The only singularity
of ζ(s) which is the simple pole at s= 1, is removed by multiplication of ζ(s) with s−1 in the definition
1Riemann [1] defines it more specially for argument s=1
2+ itand writes it ξ(t) with real tcorresponding to our
ξ1
2+ it. Our definition agrees, e.g., with Eq. (1) in Section 1.8 on p. 16 of Edwards [5] and with [20] and many
others.
5
(2.8) and the trivial zeros of ζ(s) at s=−2n, (n= 1,2, . . .) are also removed by its multiplication with
s
2!≡Γs
2+ 1which possesses simple poles there.
The functional equation
ξ(s) = ξ(1 −s),(2.12)
from which follows for the n-th derivatives
ξ(n)(s)=(−1)nξ(n)(1 −s),⇒ξ(2m+1) 1
2= 0,(n, m = 0,1,2, . . .),(2.13)
and which expresses that ξ(s) is a symmetric function with respect to s=1
2as it is immediately seen from
(2.9) and as it was first derived by Riemann [1]. It can be easily converted into the following functional
equation for the Riemann zeta function ζ(s)2
ζ(s) = (2π)s
2 (s−1)! cos πs
2ζ(1 −s).(2.14)
Together with ξ(s) = 6 (ξ(s∗))∗we find by combination with (2.12)
ξ(s) = ξ(1 −s) = (ξ(1 −s∗))∗= (ξ(s∗))∗,(2.15)
that combine in simple way, function values for 4 points (s, 1−s, 1−s∗, s∗) of the complex plane. Relation
(2.15) means that in contrast to the function ζ(s) which is only real-valued on the real axis the function ξ(s)
becomes real-valued on the real axis (s=s∗) and on the imaginary axis (s=−s∗).
As a consequence of absent zeros of the Riemann zeta function ζ(σ+ it) for σ≡Re(s)>1 together with
the functional relation (2.14) follows that all nontrivial zeros of this function have to be within the strip
0≤σ≤1 and the Riemann hypothesis asserts that all zeros of the related xi function ξ(s) are positioned
on the so-called critical line s=1
2+ it, (−∞ <t<+∞). This is, in principle, well known.
We use the functional equation (2.12) for a simplification of the notations in the following considerations
and displace the imaginary axis of the complex variable s=σ+ itfrom σ= 0 to the value σ=1
2by
introducing the entire function Ξ(z) of the complex variable z=x+ iyas follows
Ξ(z)≡ξ1
2+z, z =x+ iy=σ−1
2+ it=s−1
2,(2.16)
with the ’normalization’ (see (2.10) and (2.11))
Ξ±1
2=1
2, Ξ (0) = ξ1
2≈0.4971207782.(2.17)
following from (2.10). Thus the full relation of the Xi function Ξ(z) to the Riemann zeta function ζ(s) using
definition (2.8) is
Ξ(z) = z−1
21+2z
4!
π1+2z
4
ζ1
2+z.(2.18)
We emphasize again that the argument displacement (2.16) is made in the following only for convenience of
notations and not for some more principal reason.
The functional equation (2.12) together with (2.13) becomes
Ξ(z) = Ξ(−z), Ξ(n)(z) = (−1)nΞ(n)(−z),(2.19)
2According to Havil [10], (p. 193), already Euler correctly conjectured this relation for the zeta function ζ(s)
which is equivalent to relation (2.12) for the function ξ(s) but could not prove it. Only Riemann proved it first.
6
and taken together with the symmetry for the transition to complex conjugated variable
Ξ(z) = Ξ(−z) = (Ξ(−z∗))∗= (Ξ(z∗))∗.(2.20)
This means that the Xi function Ξ(z) becomes real-valued on the imaginary axis z= i ywhich becomes the
critical line in the new variable z
Ξ(iy) = Ξ(−iy) = (Ξ(iy))∗= (Ξ(−iy))∗.(2.21)
Furthermore, the function Ξ(z) becomes a symmetrical function and a real-valued one on the real axis z=x
Ξ(x) = Ξ(−x)=(Ξ(−x))∗= (Ξ(x))∗.(2.22)
In contrast to this the Riemann zeta function ζ(s) the function is not a real-valued function on the critical
line s=1
2+ itand is real-valued but not symmetric on the real axis. This is represented in Fig. 2.1.
(calculated with ”Mathematica 6” such as the further figures too). We see that not all of the zeros of the
10
20
30
40
50
t
-2
-1
1
2
3
ReHΖH1
2+ätLL
10
20
30
40
50
t
-2
-1
1
2
3
Re
Im HΖH1
2+ätLL
10
20
30
40
50
t
-2
-1
1
2
3
ImHΖH1
2+ätLL
0
10
20
30
40
50
t
0.5
1.0
1.5
2.0
2.5
3.0
3.5
ÈΖH1
2+ätLÈ
Figure 2.1: Real and imaginary part and absolute value of Riemann zeta function on critical line.
The position of the zeros of the whole function ζ1
2+ iton the critical line are shown by grid lines. One can see that not
all zeros of the real part are also zeros of the imaginary part and vice versa. The figures are easily to generate by program
”Mathematica” and are published in similar forms already in literature.
real part Re ζ1
2+ itare also zeros of the imaginary part Im ζ1
2+ itand, vice versa, that not all of
the zeros of the imaginary part are also zeros of the real part and thus genuine zeros of the function ζ1
2+ it
which are signified by grid lines. Between two zeros of the real part which are genuine zeros of ζ1
2+ itlies
in each case (exception first interval) an additional zero of the imaginary part, which almost coincides with
a maximum of the real part.
Using (2.9) and definition (2.16) we find the following representation of Ξ(z)
Ξ(z) = 1
2−1
4−z2Z+∞
1
dq qz+q−z
√q
∞
X
n=1
exp −πn2q2.(2.23)
7
With the substitution of the integration variable q= eu(see also (A.10) in Appendix A) representation
(2.23) is transformed to
Ξ(z)≡1
2−21
4−z2Z+∞
0
du ch (uz) e u
2
∞
X
n=1
exp −πn2e2u.(2.24)
In Appendix A we show that (2.24) can be represented as follows (see also Eq. (2) on p. 17 in [5] which
possesses a similar principal form)
Ξ(z) = Z+∞
0
du Ω (u) ch (uz),(2.25)
with the following explicit form of the function Ω(u) of the real variable u
Ω(u)≡4eu
2
∞
X
n=1
πn2e2u2πn2e2u−3exp −πn2e2u>0,(−∞ < u < +∞).(2.26)
The function Ω(u) is symmetric
Ω(u)=+Ω(−u) = Ω(|u|),(2.27)
that means it is an even function although this is not immediately seen from representation (2.26)3. We
prove this in Appendix B. Due to this symmetry, formula (2.25) can be also represented by
Ξ(z) = 1
2Z+∞
−∞
du Ω (u) ch (uz) = 1
2Z+∞
−∞
du Ω (u) euz .(2.28)
In the formulation of the right-hand side the function Ξ(z) appears as analytic continuation of the Fourier
transform of the function Ω(u) written with imaginary argument z= iyor, more generally, with substitution
z→iz0and complex z0. From this follows as inversion of the integral transformation (2.28) using (2.27)
Ω(u) = 1
πZ+∞
−∞
dy Ξ (iy) e−iuy =1
πZ+∞
−∞
dy Ξ (iy) cos (uy),(2.29)
or due to symmetry of the integrand in analogy to (2.25)
Ω(u) = 2
πZ+∞
0
dy Ξ (iy) cos (uy),(2.30)
where Ξ(iy) is a real-valued function of the variable yon the imaginary axis
Ξ(iy) = Z+∞
0
du Ω (u) cos (uy),(Ξ(iy))∗=Ξ(iy),(2.31)
due to (2.25).
A graphical representation of the function Ω(u) and of its first derivatives Ω(1)(u),(n= 1,2,3) is given
in Fig. 2.2. The function Ω(u) is monotonically decreasing for 0 ≤u < +∞due to the non-positivity of its
first derivative Ω(1)(u)≡∂ Ω(u)
∂u which explicitly is (see also Appendix A)
Ω(1)(u) = −2e u
2
∞
X
n=1
πn2e2u8πn2e2u2−30πn2e2u+ 15exp −πn2e2u
≤0,(0 ≤u < +∞),(2.32)
3It was for us for the first time and was very surprising to meet a function where its symmetry was not easily seen
from its explicit representation. However, if we substitute in (2.26) u→ −uand calculate and plot the part of Ω(u)
for u≥0 with the obtained formula then we need much more sum terms for the same accurateness than in case of
calculation with (2.26).)
8
umin =0.2373
Ω(0)=1.787
0.2 0.4 0.6 0.8 1.0
u
0.5
1.0
1.5
Ω(u)
umin =0.2373
Ω(1)(umin)=-4.922
0.2 0.4 0.6 0.8 1.0
u
-4
-2
2
Ω(1)(u)
Figure 2.2: Function Ω(u) and its first derivative Ω(1)(u),(n= 1,2,3) (see (2.25) and (2.34)).
The function Ω(u) is positive for 0 ≤u < +∞and since its first derivative Ω(1)(u) is negative for 0 < u < +∞the function
Ω(u) is monotonically decreasing on the real positive axis. It vanishes in infinity more rapidly than any exponential function
with a polynomial in the exponent.
with one relative minimum at umin = 0.237266 of depth Ω(1) (umin) = −4.92176. Moreover, it is very
important for the following that due to presence of factors exp −πn2e2uin the sum terms in (2.26) or
in (2.32) the functions Ω(u) and Ω(1)(u) and all their higher derivatives are very rapidly decreasing for
u→+∞, more rapidly than any exponential function with a polynomial of uin the argument. In this sense
the function Ω(u) is more comparable with functions of finite support which vanish from a certain u≥u0on
than with any exponentially decreasing function. From (2.27) follows immediately that the function Ω(1)(u)
is antisymmetric
Ω(1) (u) = −Ω(1) (−u) = u
|u|
∂
∂|u|Ω(|u|),⇒Ω(1) (0) = 0,(2.33)
that means it is an odd function.
It is known that smoothness and rapidness of decreasing in infinity of a function change their role in
Fourier transformations. As the Fourier transform of the smooth (infinitely continuously differentiable)
function Ω(u) the Xi function on the critical line Ξ(iy) is rapidly decreasing in infinity. Therefore it is not
easy to represent the real-valued function Ξ(iy) with its rapid oscillations under the envelope of rapid decrease
for increasing variable ygraphically in a large region of this variable y. An appropriate real amplification
envelope is seen from (2.18) to be α(y) =
1
(1+i 2y
4)!
2π1
4
√1+4y2which rises Ξ(iy) to the level of the Riemann zeta
function ζ1
2+ iton the critical line z= iy. This is shown in Fig. 2.3. The partial picture for α(y)Ξ(iy) in
Fig. 2.3. with negative part folded up is identical with the absolute value ζ1
2+ itof the Riemann zeta
function ζ(s) on the imaginary axis s=1
2+ it(fourth partial picture in Fig. 2.1).
We now give a representation of the Xi function by the derivative of the Omega function. Using ch (uz) =
1
z
∂
∂u sh (uz) one obtains from (2.25) by partial integration the following alternative representation of the
function Ξ(z)
Ξ(z) = −1
zZ+∞
0
du Ω(1) (u) sh (uz),(2.34)
that due to antisymmetry of Ω(1)(u) and sh (uz) with respect to u→ −ucan also be written
Ξ(z) = −1
2zZ+∞
−∞
du Ω(1) (u) sh (uz) = −1
2zZ+∞
−∞
du Ω(1) (u) euz .(2.35)
9
y1
zeros:
y1=14.135
y2=21.022
y3=25.011
y4=30.425
y5=32.935
y6=37.586
y7=40.919
y8=43.327
y9=48.005
y10 =49.774
10
20
30
40
50
y
0.05
0.10
0.15
XHiyL
y1=14.135
y2=21.022
5
10
15
20
25
y
-0.010
-0.005
0.005
0.010
0.015
0.020
0.025
XHiyL
y4=30.425
y5=32.935
y6=37.586
30
35
40
45
50
y
-1.2´10-7
-1.´10-7
-8.´10-8
-6.´10-8
-4.´10-8
-2.´10-8
XHiyL
ΑHyL=1
ËJ1+i2 y
4N!Ë2 Π 1
4
1+4 y2
10
20
30
40
50
y
-4
-2
2
4
ΑHyLXHiyL
Figure 2.3: Xi Function Ξ(iy) on the imaginary axis z= iy(corresponding to s=1
2+ iy).
The envelope over the oscillations of the real-valued function Ξ(iy) decreases extremely rapidly with increase of the variable y
in the shown intervals. This behavior makes it difficult to represent this function graphically for large intervals of the variable
y. By an enhancement factor which rises the amplitude to the level of the zeta function ζ(s) we may see the oscillations under
the envelope (last partial picture). A similar picture one obtains for the modulus of the Riemann zeta function
ζ1
2+ iy
only with our negative parts folded to the positive side of the abscissa, i.e. |Ξ(iy)|=
ζ1
2+ iy
(see also Fig. 2.1 (last partial
picture)). The given values for the zeros at 1
2±iynwere first calculated by J.-P. Gram in 1903 up to y15 [5]. We emphasize
here that the shown very rapid decrease of the Xi function at the beginning of yand for y→ ±∞ is due to the ’very high’
smoothness of Ω(u) for arbitrary u.
Figure 2.2 gives a graphical representation of the function Ω(u) and of its first derivative Ω(1)(u)≡∂Ω
∂u (u)
which due to rapid convergence of the sums is easily to generate by computer. One can express Ξ(z) also
by higher derivatives Ω(n)(u)≡∂nΩ
∂un(u) of the Omega function Ω(u) according to
Ξ(z) = 1
z2mZ+∞
0
du Ω(2m)(u) ch(uz)
=−1
z2m+1 Z+∞
0
du Ω(2m+1) (u) sh(uz),(m= 0,1,2, . . .),(2.36)
with the symmetries of the derivatives of the function Ω(u) for u↔ −u
Ω(2m)(u) = +Ω(2m)(−u), Ω(2m+1) (u) = −Ω(2m+1)(−u),⇒Ω(2m+1)(0) = 0,(m= 0,1, . . .).(2.37)
This can be seen by successive partial integrations in (2.25) together with complete induction. The functions
Ω(n)(u) in these integral transformations are for n≥1 not monotonic functions.
We mention yet another representation of the function Ξ(z). Using the transformations
tn≡πn2e2u,⇒dtn= 2πn2e2udu = 2tndu, du =dtn
2u,(2.38)
10
the function Ξ(z) according to (2.28) with the explicit representation of the function Ω(u) in (2.26) can now
be represented in the form
Ξ(z) =
∞
X
n=1
1
(πn2)1
42πn2−z
2Γ9
4+z
2, πn2+πn2z
2Γ9
4−z
2, πn2
−3πn2−z
2Γ5
4+z
2, πn2+πn2z
2Γ5
4−z
2, πn2,(2.39)
where Γ(α, x) denotes the incomplete Gamma function defined by (e.g., [18, 21, 36]
Γ(α, x)≡Z+∞
x
dt e−ttα−1≡Γ(α)−γ(α, x).(2.40)
However, we did not see a way to prove the Riemann hypothesis via the representation (2.39).
The Riemann hypothesis for the zeta function ζ(s=σ+ it) is now equivalent to the hypothesis that
all zeros of the related entire function Ξ(z=x+ iy) lie on the imaginary axis z= iythat means on the
line to real part x= 0 of z=x+ iywhich becomes now the critical line. Since the zeta function ζ(s) does
not possess zeros in the convergence region σ > 1 of the Euler product (2.1) and due to symmetries (2.27)
and (2.31) it is only necessary to prove that Ξ(z) does not possess zeros within the strips −1
2≤x < 0 and
0< x ≤+1
2to both sides of the imaginary axis z= iywhere for symmetry the proof for one of these strips
would be already sufficient. However, we will go another way where the restriction to these strips does not
play a role for the proof.
3. Application of second mean-value theorem of calculus to Xi
function
After having accepted the basic integral representation (2.25) of the entire function Ξ(z) according to
Ξ(z)≡Z+∞
0
du Ω(u) ch(uz),(3.1)
with the function Ω(u) explicitly given in (2.26) we concentrate us on its further treatment. However, we
do this not with this specialization for the real-valued function Ω(u) but with more general suppositions for
it. Expressed by real part U(x, y) and imaginary part V(x, y) of Ξ(z)
Ξ(x+ iy)≡U(x, y)+iV(x, y), U (x, y) = (U(x, y ))∗, V (x, y) = (V(x, y))∗,(3.2)
we find from (3.1)
U(x, y) = Z+∞
0
du Ω(u) ch(ux) cos(uy), V (x, y) = Z+∞
0
du Ω(u) sh(ux) sin(uy).(3.3)
We suppose now as necessary requirement for Ω(u) and satisfied in the special case (2.26)
Ω(u)≥0,(0 ≤u < +∞),⇒Ω(0) >0.(3.4)
Furthermore, Ξ(z) should be an entire function that requires that the integral (3.1) is finite for arbitrary
complex zand therefore that Ω(u) is rapidly decreasing in infinity, more precisely
lim
u→+∞
Ω(u)
exp (−λu)= 0,0<0≤λ < +∞,(3.5)
for arbitrary λ≥0. This means that the function Ω(u) should be a nonsingular function which is rapidly
decreasing in infinity, more rapidly than any exponential function e−λu with arbitrary λ > 0. Clearly, this
is satisfied for the special function Ω(u) in (2.26).
11
Our conjecture for a longer time was that all zeros of Ξ(z) lie on the imaginary axis z= iyfor a large
class of functions Ω(u) and that this is not very specific for the special function Ω(u) given in (2.26) but is
true for a much larger class. It seems that to this class belong all non-increasing functions Ω(u), i.e such
functions for which holds Ω(1)(u)≤0 for its first derivative and which rapidly decrease in infinity. This
means that they vanish more rapidly in infinity than any power functions |u|−n,(n= 1,2, . . .) (practically
they vanish exponentially). However, for the convergence of the integral (3.1) in the whole complex z-plane
it is necessary that the functions have to decrease in infinity also more rapidly than any exponential function
exp(−λu) with arbitrary λ > 0 expressed in (3.5). In particular, to this class belong all rapidly decreasing
functions Ω(u) which vanish from a certain u≥u0on and which may be called non-increasing finite functions
(or functions with compact support). On the other side, continuity of its derivatives Ω(n)(u),(n= 1,2, . . .)
is not required. The modified Bessel functions Iν(z) ’normalized’ to the form of entire functions 2
zνIν(z)
for ν≥1
2possess a representation of the form (3.1) with a function Ω(u) which vanishes from u≥1 on but
a number of derivatives of Ω(u) for the functions is not continuous at u= 1 depending on the index ν. It is
valuable that here an independent proof of the property that all zeros of the modified Bessel functions Iν(u)
lie on the imaginary axis can be made using their differential equations via duality relations. We intend to
present this in detail in a later work.
Furthermore, to the considered class belong all monotonically decreasing functions with the described
rapid decrease in infinity. The fine difference of the decreasing functions to the non-increasing functions Ω(u)
is that in first case the function Ω(u) cannot stay on the same level in a certain interval that means we have
Ω(1)(u)<0 for all points u > 0 instead of Ω(1)(u)≤0 only. A function which decreases not faster than e−λu
in infinity does not fall into this category as, for example, the function sech(z)≡1
ch(z)shows. On the other
side, also some simply calculable discrete superpositions such as a1ch(u) + a2ch(2u) or a1ch(z) + a3ch(3z)
as function Ξ(z) with positive amplitudes ando not provide a counterexample that the zeros lie outside the
imaginary axis but show that if the amplitudes ando not possess a definite sign then they may possess zeros
outside the imaginary axis.
To apply the second mean-value theorem it is necessary to restrict us to a class of functions Ω(u)→f(u)
which are non-increasing that means for which for all u1< u2in considered interval holds
f(a)≥f(u1)≥f(u2)≥f(b)≥0,(a≤u1≤u2≤b),(3.6)
or equivalently in more compact form
f(1)(u)≤0,(a≤u≤b).(3.7)
In case of f(1)(u) = 0 for certain uthe next higher non-vanishing derivative should be negative. The
monotonically decreasing functions in the interval a≤u≤b, in particular, belong to the class of non-
increasing functions with the fine difference that here
f(a)> f(u1)> f(u2)> f(b)>0,(a<u1< u2< b),(3.8)
is satisfied. If furthermore g(u) is a continuous function in the interval a≤u≤bthe second mean-value
theorem (often called theorem of Bonnet (1867) or Gauss-Bonnet theorem) states an equivalence for the
following integral on the left-hand side to the expression on the right-hand side according to (see some
monographs about calculus or real analysis; we recommend the monographs of Courant [37] (Appendix to
chap IV) and of Widder [38] who called it Weierstrass form of Bonnet’s theorem (chap. 5, §4))
Zb
a
du f (u)g(u) = f(a)Zu0
a
du g(u) + f(b)Zb
u0
du g(u),(a≤u0≤b),(3.9)
where u0is a certain value within the interval boundaries a < b which as a rule we do not exactly know.
It holds also for non-decreasing functions which include the monotonically increasing functions as special
class in analogous way. The proof of the second mean-value theorem is comparatively simple by applying a
substitution in the (first) mean-value theorem of integral calculus [37, 38].
12
Applied to our function f(u) = Ω(u) which in addition should rapidly decrease in infinity according to
(3.5) this means in connection with monotonic decrease that it has to be positively semi-definite if Ω(0) >0
and therefore
Ω(0) ≥Ω(u)≥0, Ω(1) (u)≤0,(0 ≤u≤+∞), Ω(u→+∞)→0,(3.10)
and the theorem (3.9) takes on the form
Z+∞
0
du Ω(u)g(u) = Ω(0) Zu0
0
du g(u),(0 ≤u0<+∞),(3.11)
where the extension to an upper boundary b→+∞in (3.9) for f(+∞) = 0 and in case of existence of the
integral is unproblematic.
If we insert in (3.9) for g(u) the function ch(uz) which apart from the real variable udepends in
parametrical way on the complex variable zand is an analytic function of zwe find that u0depends on this
complex parameter also in an analytic way as follows
Ξ(z)≡Z+∞
0
du Ω(u) ch(uz) = Ω(0) Zw0(z)
0
du ch(uz)
=Ω(0) sh (w0(z)z)
z, w0(z) = u0(x, y)+iv0(x, y),(3.12)
where w0(x+ iz) = u0(x, y) + iv0(x, y) is an entire function with u0(x, y) its real and v0(x, y) its imaginary
part. The condition for zeros z6= 0 is that sh(w0(z)z)zvanishes that leads to
w0(z)z= (u0(x, y)+iv0(x, y)) (x+ iy) = inπ, (n= 0,±1,±2, . . .),(3.13)
or split in real and imaginary part
u0(x, y)x−v0(x, y)y= 0,(3.14)
for the real part and
u0(x, y)y+v0(x, y)x=nπ, (n= 0,±1,±2, . . .),(3.15)
for the imaginary part.
The multi-valuedness of the mean-value functions in the conditions (3.13) or (3.15) is an interesting
phenomenon which is connected with the periodicity of the function g(u) = ch(uz) on the imaginary axis
z= iyin our application (3.12) of the second mean-value theorem (3.11). To our knowledge this is up to
now not well studied. We come back to this in the next Sections 4 and, in particular, Section 7 brings some
illustrative clarity when we represent the mean-value functions graphically. At present we will say only that
we can choose an arbitrary nin (3.15) which provides us the whole spectrum of zeros z1, z2, . . . on the upper
half-plane and the corresponding spectrum of zeros z−1=−z1, z−2=−z2, . . . on the lower half-plane of C
which as will be later seen lie all on the imaginary axis. Since in computer calculations the values of the
Arcus Sine function are provided in the region from −π
2to +π
2it is convenient to choose n= 0 but all other
values of nin (3.15) lead to equivalent results.
One may represent the conditions (3.14) and (3.15) also in the following equivalent form
u0(x, y) = y
x2+y2nπ, v0(x, y) = x
x2+y2nπ, (3.16)
from which follows
u2
0(x, y) + v2
0(x, y)x2+y2= (nπ)2,v0(x, y)
u0(x, y)=x
y.(3.17)
13
All these forms (3.14)–(3.17) are implicit equations with two variables (x, y) which cannot be resolved
with respect to one variable (e.g., in forms y=yk(x) for each fixed nand branches k) and do not provide
immediately the necessary conditions for zeros in explicit form but we can check that (3.16) satisfies the
Cauchy-Riemann equations as a minimum requirement
∂u0(x, y)
∂x =∂v0(x, y)
∂y ,∂u0(x, y)
∂y =−∂v0(x, y)
∂x .(3.18)
We have to establish now closer relations between real and imaginary part u0(x, y) and v0(x, y) of the complex
mean-value parameter w0(z=x+ iy). The first step in preparation to this aim is the consideration of the
derived conditions on the imaginary axis.
4. Specialization of second mean-value theorem to Xi function on
imaginary axis
By restriction to the real axis y= 0 we find from (3.3) for the function Ξ(z)
Ξ(x) = U(x, 0), V (x, 0) = 0,(4.1)
with the following two possible representations of U(x, 0) related by partial integration
U(x, 0) = Z+∞
0
du Ω(u) ch(ux) = −1
xZ+∞
0
du Ω(1) (u) sh(ux)>0.(4.2)
The inequality U(x, 0) >0 follows according to the supposition Ω(u)≥0, Ω(0) >0 from the non-negativity
of the integrand that means from Ω(u) ch(ux)≥0. Therefore, the case y= 0 can be excluded from the
beginning in the further considerations for zeros of U(x, y) and V(x, y ).
We now restrict us to the imaginary axis x= 0 and find from (3.3) for the function Ξ(z)
Ξ(iy) = U(0, y), V (0, y) = 0.(4.3)
with the following two possible representations of U(0, y) related by partial integration
U(0, y) = Z+∞
0
du Ω(u) cos(uy) = −1
yZ+∞
0
du Ω(1) (u) sin(uy).(4.4)
From the obvious inequality
−1≤cos(uy)≤+1,(4.5)
together with the supposed positivity of Ω(u) one derives from the first representation of U(0, y) in (4.4)
the inequality
−Ω0≤U(0, y)≤+Ω0, Ω0=U(0,0) ≡Z+∞
0
du Ω(u)≥0.(4.6)
In the same way by the inequality
−1≤sin(uy)≤+1,(4.7)
one derives using the non-positivity of Ω(1)(u) (see (3.10)) together with the second representation of U(0, y)
in (4.4) the inequality
−Ω(0) ≤U(0, y)y≤+Ω(0), Ω(0) = −Z+∞
0
du Ω(1) (u)≥0.(4.8)
14
which as it is easily seen does not depend on the sign of y. Therefore we have two non-negative parameters,
the zeroth moment Ω0and the value Ω(0), which according to (4.6) and (4.8) restrict the range of values of
U(0, y) to an interior range both to (4.6) and to (4.8) at once.
For mentioned purpose we now consider the restriction of the mean-value parameter w0(z) to the imagi-
nary axis z= iyfor which g(u) = ch(u(iy)) = cos(uy) is a real-valued function of y. For arbitrary fixed ywe
find by the second mean-value theorem a parameter u0in the interval 0 ≤y < +∞which naturally depends
on the chosen value ythat means u0=u0(0, y). The extension from the imaginary axis z= iyto the whole
complex plane Ccan be made then using methods of complex analysis. We discuss some formal approaches
to this in Appendix B. Now we apply (3.12) to the imaginary axis z= iy.
The second mean-value theorem (3.12) on the imaginary axis z= iy(or x= 0) takes on the form
Z+∞
0
du Ω(u) ch(u(iy)) = Z+∞
0
du Ω(u) cos(uy) = Ω(0) Zu0(0,y)
0
du cos(uy)
=Ω(0)sin (u0(0, y)y)
y,(u0(0, y)6= 0, v0(0, y) = 0) .(4.9)
As already said since the left-hand side is a real-valued function the right-hand side has also to be real-valued
and the parameter function w0(iy) is real-valued and therefore it can only be the real part u0(0, y) of the
complex function w0(z=x+ iy) = u0(x, y)+iv0(x, y) for x= 0.
The second mean-value theorem states that u0(0, y) lies between the minimal and maximal values of the
integration borders that is here between 0 and +∞and this means that u0(0, y) should be positive. Here
arises a problem which is connected with the periodicity of the function g(u) = cos(uy) as function of the
variable ufor fixed variable yin the application of the mean-value theorem. Let us first consider the special
case y= 0 in (4.9) which leads to
Z+∞
0
du Ω(u) = Ω(0) Zu0
0
du =Ω(0)u0=Ω(0)u0lim
y→0
sin(u0y)
u0y
| {z }
= 1
, u0≡u0(0,0) >0.(4.10)
From this relation follows u0≡u0(0,0) >0 and it seems that all is correct also with the continuation to
u0(0, y)>0 for arbitrary y. One may even give the approximate values Ω(0) ≈1.78679 and u0≈0.27822
and therefore Ω0≡Ω(0)u0≈0.49712 which, however, are not of importance for the later proofs. If we now
start from u0(0,0) >0 and continue it continuously to u0(0, y) then we see that u0(0, y) goes monotonically
to zero and approaches zero approximately at y=y1≈14.135 that is at the first zero of the function Ξ(iy)
on the positive imaginary axis and goes then first beyond zero and oscillates then with decreasing amplitude
for increasing yaround the value zero with intersecting it exactly at the zeros of Ξ(iy). We try to illustrate
this graphically in Section 7. All zeros lie then on the branch u0(0, y)y=nπ with n= 0. That u0(0, y)
goes beyond zero seems to contradict the content of the second mean-value theorem according which u0(0, y)
has to be positive in our application. Here comes into play the multi-valuedness of the mean-value function
u0(0, y). For the zeros of sin(u0(0, y)y) in (4.9) the relations u0(0, y)y=nπ with different integers nare
equivalent and one may find to values u0(0, y)<0 equivalent curves u0(n; 0, y) with u0(n; 0, y)>0 and all
these curves begin with u0(n6= 0; 0,0) → ∞ for y→0. However, we cannot continue u0(0,0) in continuous
way to only positive values for u0(0, y).
For |y|→∞the inequality (4.8) is stronger than (4.6) and characterizes the restrictions of U(0, y) and
via the equivalence U(0, y)y=Ω(0) sin(u0(0, y)y) follows from (4.8)
n−1
2π≤u0(0, y)y= arcsin U(0, y )y
Ω(0) ≤n+1
2π, (n= 0,±1,±2, . . .),(4.11)
where the choice of ndetermines a basis interval of the involved multi-valued function arcsin(z) and the
inequality says that it is in every case possible to choose it from the same interval of length π. The zeros
ykof the Xi function Ξ(x+ iy) on the imaginary axis x= 0 (critical line) are determined alone by the
15
(multi-valued) function u0(0, y) whereas v0(0, y) vanishes automatically on the imaginary axis in considered
special case and does not add a second condition. Therefore, the zeros are the solutions of the conditions
u0(0, y)y=nπ, (n= 0,±1,±2, . . .),(v0(0, y ) = 0).(4.12)
It is, in general, not possible to obtain the zeros ykon the critical line exactly from the mean-value function
u0(0, y) in (4.9) since generally we do not possess it explicitly.
In special cases the function u0(0, y) can be calculated explicitly that is the case, for example, for all
(modified) Bessel functions 2
zνIν(z). The most simple case among these is the case ν=1
2when the
corresponding function Ω(u) is a step function
Ω(u) = Ω(0)θ(u0−u),(4.13)
where θ(x) = 0, x < 0
1, x > 0is the Heaviside step function. In this case follows
Ξ(z) = Ω(0) Zu0
0
du ch(uz) = Ω(0)u0
sh(u0z)
u0z=Ω(0)u0
| {z }
=Ω0π
2u0z1
2
I1
2(u0z),(4.14)
where Ω(0)u0=R+∞
0du Ω(u) is the area under the function Ω(u) = Ω(0)θ(u0−u) (or the zeroth-order
moment of this function. For the squared modulus of the function Ξ(z) we find
Ξ(z) (Ξ(z))∗= (Ω(0))2sh2(u0x) + sin2(u0y)
x2+y2= (Ω(0))2ch(2u0x)−cos(2u0y)
2 (x2+y2),(4.15)
from which, in particular, it is easy to see that this special function Ξ(x+ iy) possesses zeros only on the
imaginary axis z= iyor x= 0 and that they are determined by
u0yn=nπ, ⇒yn=nπ
u0
,(n=±1,±2, . . .).(4.16)
The zeros on the imaginary axis are here equidistant but the solution y0= 0 is absent since then also the
denominators in (4.15) are vanishing. The parameter w0(z) in the second mean-value theorem is here a real
constant u0in the whole complex plane
w0(x+ iy) = u0(x, y)+iv0(x, y) = u0,⇒
u0(x, y) = u(0, y) = u0, v0(x, y ) = v0(0, y) = 0.(4.17)
Practically, the second mean-value theorem compares the result for an arbitrary function Ω(u) under the
given restrictions with that for a step function Ω(u) = Ω(0) θ(u0−u) by preserving the value Ω(0) and
making the parameter u0depending on zin the whole complex plane. Without discussing now quantitative
relations the formulae (4.17) suggest that v0(x, y) will stay a ’small’ function compared with u0(x, y) in the
neighborhood of the imaginary axis (i.e. for |x| |y|) in a certain sense.
We will see in next Section that the function u0(0, y) taking into account v0(0, y) = 0 determines the
functions u0(x, y) and v0(x, y) and thus w0(z) in the whole complex plane via the Cauchy-Riemann equations
in an operational approach that means in an integrated form which we did not found up to now in literature.
The general formal part is again delegated to an Appendix B.
5. Accomplishment of proof for zeros of Xi functions on imaginary
axis alone
In last Section we discussed the application of the second mean-value theorem to the function Ξ(z) on the
imaginary axis z= iy. Equations (3.14) and (3.15) or their equivalent forms (3.16) or (3.17) are not yet
16
sufficient to derive conclusions about the position of the zeros on the imaginary axis in dependence on x6= 0.
We have yet to derive more information about the mean-value functions w0(z) which we obtain by relating
the real-valued function u0(x, y) and v0(x, y) to the function u0(0, y) on the imaginary axis (v0(0, y) = 0).
The general case of complex zcan be obtained from the special case z= iyin (4.9) by application of the
displacement operator exp −ix∂
∂y to the function Ξ(iy) according to
Ξ(x+ iy) = exp −ix∂
∂y Ξ(iy) exp ix∂
∂y = exp −ix∂
∂y Z+∞
0
du Ω(u) ch(iuy) exp ix∂
∂y
= exp −ix∂
∂y Ω(0)
shiu0(0, y)y
iyexp ix∂
∂y =Ω(0)
shu0(0, y −ix)i(y−ix)
i(y−ix).(5.1)
The function u0(0, y −ix) = w0(x+ iy) = u0(x, y)+iv0(x, y ) is related to u0(0, y) as follows
u0(x, y) = cos x∂
∂y u0(0, y), v0(x, y) = −sin x∂
∂y u0(0, y),(5.2)
or in more compact form
w0(x+ iy) = exp −ix∂
∂y u0(0, y) = u0(0, y −ix).(5.3)
This is presented in Appendix B in more general form for additionally non-vanishing v0(0, y) and arbi-
trary holomorphic functions. It means that we may obtain u0(x, y) and v0(x, y) by applying the operators
cos x∂
∂y and −sin x∂
∂y , respectively, to the function u0(0, y) on the imaginary axis (remind v0(0, y)=0
vanishes there in our case). Clearly, equations (5.2) are in agreement with the Cauchy-Riemann equations
∂u0
∂x =∂ v0
∂y and ∂ u0
∂y =−∂ v0
∂x as a minimal requirement.
We now write Ξ(z) in the form equivalent to (5.1)
Ξ(x+ iy) = Ω(0)
shu0(x, y)+iv0(x, y)(x+ iy)
x+ iy
=Ω(0)
shu0(x, y)x−v0(x, y)y+ iu0(x, y )y+v0(x, y)x
x+ iy.(5.4)
The denominator x+ iydoes not contribute to zeros. Since the Hyperbolic Sine possesses zeros only on the
imaginary axis we see from (5.4) that we may expect zeros only for such related variables (x, y) which satisfy
the necessary condition of vanishing of its real part of the argument that leads as we already know to (see
(3.14))
u0(x, y)x−v0(x, y)y= 0.(5.5)
The zeros with coordinates (xk, yk) themselves can be found then as the (in general non-degenerate) solutions
of the following equation (see (3.15))
u0(x, y)y+v0(x, y)x=nπ, (n= 0,±1,±2, . . .),(5.6)
if these pairs (x, y) satisfy the necessary condition (5.5). Later we will see provides the whole spectrum of
solutions for the zeros but we can also obtain each (xk, yk) separately from one branch nand would they
then denote by (xn, yn). Thus we have first of all to look for such pairs (x, y) which satisfy the condition
(5.5) off the imaginary axis that is for x6= 0 since we know already that these functions may possess zeros
on the imaginary axis z= iy.
17
Using (5.2) we may represent the necessary condition (5.5) for the proof by the second mean-value
theorem in the form
xcos x∂
∂y u0(0, y) + ysin x∂
∂y u0(0, y)=0,(5.7)
and equation (5.6) which determines then the position of the zeros can be written with equivalent values n
ycos x∂
∂y u0(0, y)−xsin x∂
∂y u0(0, y) = nπ, (n= 0,±1,±2, . . .).(5.8)
We may represent Eqs. (5.7) and (5.8) in a simpler form using the following operational identities
xcos x∂
∂y +ysin x∂
∂y = sin x∂
∂y y,
ycos x∂
∂y −xsin x∂
∂y = cos x∂
∂y y, (5.9)
which are a specialization of the operational identities (B.11) in Appendix B with w(z) = u(x, y)+iv(x, y)→
z=x+iyand therefore u(x, y)→x, v(x, y)→y. If we multiply (5.7) and (5.8) both by the function u0(0, y)
then we may write (5.7) in the form (changing order yu0(0, y) = u0(0, y)y)
sin x∂
∂y (u0(0, y)y)=0,(5.10)
and (5.8) in the form
cos x∂
∂y (u0(0, y)y) = nπ, (n= 0,±1,±2, . . .).(5.11)
The left-hand side of these conditions possess the general form for the extension of a holomorphic function
W(z) = U(x, y) +iV(x, y) from the functions U(0, y) and V(0, y) on the imaginary axis to the whole complex
plane in case of V(0, y) = 0 and if we apply this to the function U(0, y) = u0(0, y)y. Equations (5.10) and
(5.11) possess now the most simple form, we found, to accomplish the proof for the exclusive position of
zeros on the imaginary axis. All information about the zeros of the Xi function Ξ(z) = U(x, y )+iV(x, y)
for arbitrary xis now contained in the conditions (5.10) and (5.11) which we now discuss.
Since cos x∂
∂y is a nonsingular operator we can multiply both sides of equation (5.11) by the
inverse operator cos−1x∂
∂y and obtain
u0(0, y)y= cos−1x∂
∂y nπ =(1 + 1
2x∂
∂y 2
+. . .)nπ =nπ, (n= 0,±1, . . .).(5.12)
This equation is yet fully equivalent to (5.11) for arbitrary xbut it provides only the same solutions
for the values yof zeros as for zeros on the imaginary axis. This alone already suggests that it
cannot be that zeros with x6= 0 if they exist possess the same values of yas the zeros on the
imaginary axis. But in such form the proof of the impossibility of zeros off the imaginary axis
seemed to be not satisfactory and we present in the following some slightly different variants which
go deeper into the details of the proof.
In analogous way by multiplication of (5.10) with the operator sin x∂
∂y and (5.11) with the
operator cos x∂
∂y and addition of both equations we also obtain condition (5.12) that means
u0(0, y)y=sin2x∂
∂y + cos2x∂
∂y u0(0, y)y=nπ, (n= 0,±1,±2, . . .),(5.13)
18
The equal conditions (5.12) and (5.13) which are identical with the condition for zeros on the
imaginary axis are a necessary condition for all zeros. For each chosen equivalent n(remind
u0(0, y) depends then on nwhich we do not mention by the notation) one obtains an infinite series
of solutions ykfor the zeros of the function Ξ(iy)
u0(0, yk)yk=nπ, {u0(0, y)y}y6=yk6=nπ, (5.14)
whereas for y6=ykequation (5.12), by definition of yk, is not satisfied. Supposing that we know
u0(0, y) that is as a rule not the case, we could solve for each n= 0,±1,±2, . . . the usually tran-
scendental equations (5.13) graphically, for example, by drawing the equivalent functions u0(0, y)y
over variable yas abscissa and looking for the intersections points with the lines nπ over y(Section
7). These intersection points y=ykare the solutions for zeros ykon the imaginary axis. Choosing
x= 0 the second condition (5.10) is identically satisfied.
Now we have to look for additional zeros (x, yk) with x6= 0. Whereas for zeros with x= 0 the
condition (5.10) is identically satisfied we have to examine this condition for zeros with x6= 0. In
the case of x6= 0 we may divide both sides of the condition (5.10) by xand obtain
sin x∂
∂y
x(u0(0, y)y) =
sin x∂
∂y
x∂
∂y
∂
∂y (u0(0, y)y) = 0.(5.15)
Since sinx∂
∂y
x∂
∂y
is a nonsingular operator (in contrast to sin x∂
∂y which possesses 0 as eigenvalue
to eigenfunction f(x, y) = constxn,(n= 0,1, . . .)) we may multiply equation (5.15) by the inverse
operator x∂
∂y
sinx∂
∂y and obtain
∂
∂y (u0(0, y)y) = ∂u0
∂y (0, y)y+u0(0, y) = 0.(5.16)
This condition has also to be satisfied for the solution y=ykof (5.12) which make this equation
to the identity (5.14) that means that
∂u0
∂y (0, yk)yk+u0(0, yk) = ∂u0
∂y (0, yk)yk+nπ
yk
= 0,(5.17)
has to be identically satisfied. Moreover, if we apply the operator sinmx∂
∂y
xm+1 to condition (5.10) we
obtain
sinm+1 x∂
∂y
xm+1 (u0(0, y)y) =
sin x∂
∂y
x∂
∂y
m+1
∂m+1
∂ym+1 (u0(0, y)y) = 0,(m= 0,1,2, . . .),(5.18)
and by multiplication with the inverse operator to the nonsingular operator x∂
∂y
sinx∂
∂y !m+1
we
find
∂m+1
∂ym+1 (u0(0, y)y) = 0,(m= 0,1,2, . . .).(5.19)
19
All these conditions have to be satisfied for the solutions y=ykwith x6= 0 in (5.13), i.e.
∂m+1
∂ym+1 {(u0(0, y)y)}y=yk= 0,(m= 0,1,2, . . .).(5.20)
The same conditions follow also from (5.11) combined with (5.11) by Taylor series expansion with
respect to x∂
∂y for x6= 0 according to
nπ = exp ±ix∂
∂y (u0(0, y)y) = u0(0, y)y+∞
X
m=0
(±ix)m+1
(m+ 1)!
∂m+1
∂ym+1 (u0(0, y)y),(5.21)
by setting all sum terms proportional to xm+1 equal to zero.
If we make now a Taylor series expansion of the function u0(0, y)yin the neighborhood y=yk
of a solution which obeys all conditions (5.14) and (5.20) then we find
u0(0, y)y=u0(0, yk)yk+∞
X
m=0
1
(m+ 1)!
∂m+1
∂ym+1 {(u0(0, y)y)}y=yk(y−yk)m+1 =nπ. (5.22)
Thus we can find zeros for x6= 0 that means off the imaginary axis if the mean value function
u0(0, y) on the imaginary axis possess one of the forms
u0(0, y)y=nπ, ⇔u0(0, y) = nπ
y,(n= 0,±1,±2, . . .),(5.23)
for a certain integer n. According to (5.2) the whole mean-value functions u0(x, y) and v0(x, y) are
then
u0(x, y) = cos x∂
∂y nπ
y=nπ
2exp ix∂
∂y + exp −ix∂
∂y 1
y
=nπ
21
y+ ix+1
y−ix=nπ y
x2+y2,
v0(x, y) = −sin x∂
∂y kπ
y=−nπ
i2 exp ix∂
∂y −exp −ix∂
∂y 1
y
=−nπ
i2 1
y+ ix−1
y−ix=nπ x
x2+y2,(5.24)
or in compact form
w0(z) = u0(x, y)+iv0(x, y) = nπ y+ ix
x2+y2= i nπ
x+ iy,⇔w0(z)z= inπ. (5.25)
If we insert w0(z)z= inπ into equation (3.12) then we get Ξ(z) = 0 for all n= 0,±1,±2, . . ..
This means that all conditions for zeros with x6= 0 together do not lead to a solution for arbitrary
Ξ(z)6= 0. Thus we have proved that all zeros of Xi functions Ξ(z) lie on the imaginary axis x= 0.
Recognizing equations (5.10) and (5.11) as correct ones one may find modifications of the given
proof of impossibility of solutions for zeros with x6= 0. For example, one can make Taylor series
expansions and obtain from equation (5.11)
u0(0, y)y+∞
X
m=1
(−1)mx2m
(2m)!
∂2m
∂y2m(u0(0, y)y) = nπ, (n= 0,±1,±2, . . .),(5.26)
20
and from (5.10)
∞
X
m=0
(−1)mx2m+1
(2m+ 1)!
∂2m+1
∂y2m+1 (u0(0, y)y)=0.(5.27)
The right-hand sides of these equations are independent of variable xand therefore the left-hand
sides must it too. On the imaginary axis for x= 0 this leads to the condition (5.13) as the only
condition which determines the zeros on this axis. For x6= 0 one has as additional condition the
vanishing of the coefficients in front of powers of xn+1,(n= 0,1,2, . . .) in (5.26) and (5.27) that
taken together leads to the conditions (5.19). The further discussion of the impossibility of this
case is the same as before.
A third variant to show the impossibility for zeros in case of x6= 0 is to make the transition
to the Fourier transform of u0(0, y)yand to solve the equation arising from 5.15) by generalized
functions and then making the inverse Fourier transformation. One may show then that this is not
compatible with the general solution of (5.11) which determines the position of the zeros. We do
not present this variant here.
We have now finally proved that all Xi functions Ξ(z) of the form (3.1) for which the second
mean-value theorem is applicable (function Ω(u) positively semi-definite and non-increasing (or
also negatively semi-definite and non-decreasing) may possess zeros only on the imaginary axis.
6. Consequences for proof of the Riemann hypothesis
The given proof for zeros only on the imaginary axis x= 0 for the considered Xi function Ξ(z) =
Ξ(x+ iy) includes as special case the function Ω(u) to the Riemann hypothesis which is given in
(2.26). However, it includes also the whole class of modified Bessel functions of imaginary argument
Iν(z) which possess zeros only on the imaginary axis and if we make the substitution z↔izalso
the usual Bessel function Jν(z) which possess zeros only on the real axis.
We may ask about possible degeneracies of the zeros of the Xi functions Ξ(z) on the imaginary
axis z= iy. Our proof does not give a recipe to see whether such degeneracies are possible or
not. In case of the Riemann zeta function Ξ(z)↔ξ(s)↔ζ(s) one cannot expect a degeneracy
because the countable number of all nontrivial zeros are (likely) irrational (transcendental, proof?)
numbers but we do not know a proof for this.
For Ξ(z) as an entire function one may pose the question of its factorization with factors of
the form 1 −z
znwhere zngoes through all roots where in case of degeneracy the same factors are
taken multiple times according to the degeneracy. It is well known that an entire function using
its ordered zeros zn,(|zn|≤|zn+1|) can be represented in Weierstrass product form multiplied by
an exponential function eh(z)with an entire function function h(z) in the exponent with the result
that eh(z)is an entire function without zeros. This possesses the form (e.g., [15])
Ξ(z)=eh(z)zmY
n1−z
znexp Pknz
zn,(6.1)
with a polynomial Pk(w) of degree kwhich depending on the roots znmust be appropriately chosen
to guarantee the convergence of the product. This polynomial is defined by first ksum terms in
21
the Taylor series for −log(1 −w)4
Pk(w)≡w+w2
2+w3
3+. . . +wk
k=−log(1 −w)−∞
X
l=k+1
wl
l.(6.2)
By means of these polynomials the Weierstrass factors are defined as the functions
Ek(w)≡(1 −w) exp (Pk(w)) ,(6.3)
from which follows
log (Ek(w)) = log(1 −w)− log(1 −w) + ∞
X
l=k+1
wl
l!=−∞
X
l=k+1
wl
l.(6.4)
From this form it is seen that Ek(w) possesses the following initial terms of the Taylor series
Ek(w) = (1 −w) exp −∞
X
l=k+1
wl
l!= 1 −wk+1
k+ 1 +. . . , (6.5)
and is a function with a zero at w= 1 but with a Taylor series expansion which begins with the
terms 1 −wk+1
k+1 . Hadamard made a precision of the Weierstrass product form by connecting the
degree knof the polynomials in (6.1) with the order ρof growth of the entire function and showed
that kncan be chosen independently of the n-th root znby kn→k≥ρ−1 The order of Ξ(z)
which is equal to 1 is not a strict order ρ= 1 (for this last notion see [15]). However, this does not
play a role in the Hadamard product representation of Ξ(z) and the polynomials Pkn(w) in (6.1)
can be chosen as P0(w) that means equal to 0 according to kn=k=ρ−1. The entire function
h(z) in the exponent in (6.1) can be only a constant since in other case it would introduce a higher
growth of Ξ(z). Thus the product representation of Ξ(z) possesses the form
Ξ(z) = Ξ(0)Y0+∞
n=−∞ 1−z
zn=Z+∞
0
du Ω(u)
+∞
Y
n=1 1−z2
z2
n
=Ω0
+∞
Y
n=1 1 + z2
y2
n, Ξ(0) = Ω0≡Z+∞
0
du Ω(u) = ξ1
2= 0.49712... , (6.6)
where we took into account the symmetry z−n=−zn= (zn)∗of the zeros and the proof zn= iyn
that all zeros lie on the imaginary axis and a zero z0= 0 is absent. With Ω0we denoted the first
moment of the function Ω(u).
Formula (6.6) in connection with his hypothesis was already used by Riemann in [1] and later
proved by von Mangoldt where the product representation of entire functions by Weierstrass which
was later stated more precisely by Hadamard plays a role. There is another formula for an approx-
imation to the number of nontrivial zeros of ζ(s) or ξ(s) which in application to the number of
zeros N(Y) of Ξ(z) on the imaginary axis z= iy(critical line) in the interval between y= 0 and
y=Y. It takes on the form (Yfor Ξ(z) is equivalent to usual Tfor ζ(s))
N(Y) = ZY
0
dy ν(y)≈Y
2πlog Y
2π−Y
2π,(Y0),(6.7)
4Sometimes our Pk−1(w) is denoted by Pk(w).
22
with the logarithmically growing density
ν(y)≈1
2πlog y
2π,(y1).(6.8)
As long as the Riemann hypothesis was not proved it was formulated for the critical strip 0 ≤σ≤1
of the complex coordinate s=σ+ itin ξ(s) parallel to the imaginary axis and with tbetween t= 0
and t=T(with Tequal to our Yin (6.7)). It was already suggested by Riemann [1] but not proved
in detail there and was later proved by von Mangoldt in 1905. A detailed proof by means of the
argument principle can be found in [12]. The result of Hardy (1914) (cited in [5]) that there exist
an infinite number of zeros on the critical line is a step to the full proof of the Riemann hypothesis.
Section 4 of present article may be considered as involving such proof of this last statement.
We have now proved that functions Ξ(z) defined by integrals of the form (3.1) with non-
increasing functions Ω(u) which decrease in infinity sufficiently rapidly in a way that Ξ(z) becomes
an entire function of zpossess zeros only on the imaginary axis z= iy. This did not provide a
recipe to see in which cases all zeros on the imaginary axis are simple zeros but it is unlikely that
within a countable sequence of irregularly chosen (probably) transcendental numbers (the zeros)
two of them are coincident (it seems to be difficult to formulate last statement in a more rigorous
way). It also did not provide a direct formula for the number of zeros in an interval [(0,0),(0, Y )]
from zero to Yon the imaginary axis or of its density there but, as already said, Riemann [1]
suggested for this an approximate formula and von Mangoldt proved it
The proof of the Riemann hypothesis is included as the special case (2.26) of the function
Ω(u) into a wider class of functions with an integral representation of the form (3.1) which under
the discussed necessary conditions allowing the application of the second mean-value theorem of
calculus possess zeros only on the imaginary axis. The equivalent forms (2.35) and (2.36) of the
integral (3.1) where the functions, for example Ω(1)(u), are no more generally non-increasing suggest
that conditions for zeros only on the imaginary axis are existent for more general cases than such
prescribed by the second mean-value theorem. A certain difference may happen, for example, for
z= 0 because powers of it are in the denominators in the representations in (2.36).
7. Graphical illustration of mean-value parameters to Xi function
for the Riemann hypothesis
To get an imagination how the mean-value function w0(z) = w0(x+iy) looks like we calculate it for
the imaginary axis and for the real axis for the case of the function Ω(u) in (2.26) that is possible
numerically. From the two equations for general zand for z= 0
Ω(0) sh (w0(z)z)
z=Z+∞
0
du Ω(u) ch(uz),
Ω(0)w0(0) = Z+∞
0
du Ω(u),(7.1)
follows
w0(z) = 1
zArsh z
Ω(0) Z+∞
0
du Ω(u) ch(uz),
w0(0) = 1
Ω(0) Z+∞
0
du Ω(u),(7.2)
23
u0H0,yLu0H0,0L
y1=14.135
y2=21.022
y3
5
10
15
20
25
y
0.5
1.0
1.5
2.0
w0Hi yLw0H0L
Mean-value function u0H0,yLon y-axis; Hv0H0,yL=0L
u0Hx,0Lu0H0,0L
5
10
15
20
25
x
0.5
1.0
1.5
2.0
w0HxLw0H0L
Mean-value function u0Hx,0Lon x-axis; Hv0Hx,0L=0L
Figure 7.4: Mean value parameters w0(iy) = u0(0, y) and w0(x) = u0(x, 0) for the Xi function in the proof of the
Riemann hypothesis.
It is not to see in the chosen scale that the curve u0(0, y) goes beyond the y-axis and oscillates around it due to extremely
rapid vanishing of the envelope of u0(0, y) with increasing ybut we do not resolves this here by additional graphics because
this behavior is better to see in the later considered case of modified Bessel functions. Using (7.2) we calculate numerically
w0(0) = u0(0,0) ≈0.27822 that is the value which we call the optimal value for the moment series expansion (see Section 11).
The part in the second partial figure which at the first glance looks like a straight line as asymptote is not such.
with the two initial terms of the Taylor series
w0(z) = 1
z(z
Ω(0) Z+∞
0
du Ω(u) ch(uz)−1
6z
Ω(0) Z+∞
0
du Ω(u) ch(uz)3
+. . .),(7.3)
and with the two initial terms of the asymptotic series
w0(z) = 1
z(log 2z
Ω(0) Z+∞
0
du Ω(u) ch(uz)+1
4z
Ω(0) Z+∞
0
du Ω(u) ch(uz)−2
+. . .).(7.4)
From (7.2) follows
w0(z)
w0(0) =Ω(0)
zR+∞
0du Ω(u)Arsh z
Ω(0) Z+∞
0
du Ω(u) ch(uz).(7.5)
This can be numerically calculated from the explicit form (2.26) of Ω(u). For y= 0 and for x= 0
(and only for these cases) the function w0(z)
w0(0) is real-valued, in particular, for y= 0
w0(x)
w0(0) =Ω(0)
xR+∞
0du Ω(u)Arsh x
Ω(0) Z+∞
0
du Ω(u) ch(ux)
=R+∞
0du Ω(u) ch(ux)
R+∞
0du Ω(u)(1−1
6x
Ω(u)Z+∞
0
du Ω(u) ch(ux)2
+. . .)
=u0(x, 0)
u0(0,0) ,(7.6)
and for x= 0
w0(iy)
w0(0) =Ω(0)
yR+∞
0du Ω(u)arcsin y
Ω(0) Z+∞
0
du Ω(u) cos(uy)
24
n=0
n=1
n=2
n=3
w0Hi yLw0H0L=u0H0,yLu0H0,0L
w0H0L=u0H0,0L»0.27822
5
10
15
20
25
y
2
4
6
8
w0Hi yLw0H0L
Equivalent mean-value functions to u0H0,yLon y-axis
n=0
n=1
n=2
n=3
y1=14.135
y2=21.022
y3
0
5
10
15
20
25
y
Π
2
Π
3Π
2
2Π
5Π
2
3Π
u0H0,yLy
Equivalent functions u0H0,yLy
Figure 7.5: Mean value parameters in the proof of the Riemann hypothesis.
On the left-hand side there are shown the mean value parameters w0(iy)
w(0) =u0(0,y)
u0(0,0) for the Xi function to the Riemann hypothesis
if we do not take the values of the function arcsin(t) in the basic range −π
2≤t≤π
2but in equivalent ranges according to
(7.8). On the right-hand side are shown the corresponding functions u0(0, y)ywhich according to cos(x+nπ) = (−1)nsin(x)
and the condition for zeros cos (u0(0, y)y) = 0 lead to equivalent ranges kπ =u0(0, y)y∼
=u0(0, y)y+nπ = (k+n)π, (k=
±1,±2,...;n= 0,±1,±2,...) (see (4.12)) determine the zeros of the Xi function on the imaginary axis. We see that the multi-
valuedness of the arcsin(x) function does not spoil a unique result for the zeros because every branch find the corresponding n
of nπ where then all zeros lie. Due to extremely rapid decrease of the function u0(0, y ) with increasing ythis is difficult to see
(position of first three zero at y1≈14.1, y2≈21.0, y3≈25.0 is shown) but if we separate small intervals of yand enlarge the
range of values for y0(0, y) this becomes visible (similar as in Fig. 2.3). We do not make this here because this effect is better
visible for the modified Bessel functions which we intend to consider at another place.
=R+∞
0du Ω(u) cos(uy)
R+∞
0du Ω(u)(1 + 1
6y
Ω(u)Z+∞
0
du Ω(u) cos(uy)2
+. . .)
=u0(0, y)
u0(0,0),(7.7)
where we applied the first two terms of the Taylor series expansion of arcsin(x) in powers of
x. A small problem is here that we get the value for this multi-valued function in the range
−π
2≤arcsin(x)≤+π
2. Since 1
xarcsin(x) is an even function with only positive coefficients in its
Taylor series the term in braces is in every case positive that becomes important below.
The two curves which we get for w0(x)
w0(0) and for w0(iy)
w0(0) are shown in Fig. 7.4. The function
for w0(x, 0) on the real axis y= 0 (second partial picture) is not very exciting. The necessary
condition xu0(x, 0) = 0 (see (5.5)) can be satisfied only for x= 0 but it is easily to see from
Ξ(0) = R+∞
0du Ω(u)≈1.7868 6= 0 that there is no zero. For the function w0(0, y) on the
imaginary axis x= 0 the necessary condition yv0(0, y) = 0 (see (5.5)) is trivially satisfied since
v0(0, y) = 0 and does not restrict the solutions for zeros. In this case only the sufficient condition
yu0(0, y) = nπ, (n= 0,±1,±2, . . .) determines the position of the zeros on the imaginary axis.
The first two pairs of zeros are at y1≈ ±14.135, y2≈ ±21.022 and the reason that we do not
see them in Fig. 7.4 is the rapid decrease of the function u0(0, y) with increasing y. If we enlarge
this range we see that the curve goes beyond the y-axis after the first root at 14.135 of the Xi
function. As a surprise for the second mean-value method we see that the parameter u0(0, y)
becomes oscillating around this axis. This means that the roots which are generally determined by
the equation yu0(0, y) = nπ (see (5.6)) are determined here by the value n= 0 alone. The reason
25
for this is the multi-valuedness of the ArcSine function according to
arcsin(x)∼
=nπ + (−1)narcsin(x),−1≤x≤+1,−π
2≤arcsin(x)≤+π
2, n ∈Z.(7.8)
If we choose the values for the arcsin(x)-function not in the basic interval −π
2≤x≤+π
2for which
the Taylor series provides the values but from other equivalent intervals according to (7.8) we get
other curves for u0(0, y) and yu0(0, y) from which we also may determine the zeros (see Fig. 7.5),
however, with other values nin the relation yu0(0, y) = nπ, (n= 0,±1,±2, . . .) and the results
are invariant with respect to the multi-valuedness. This is better to see in case of the modified
Bessel functions for which the curves vanish less rapidly with increasing yas we intend to show at
another place. All these considerations do not touch the proof of the non-existence of roots off the
imaginary axis but should serve only for better understanding of the involved functions. It seems
that the specific phenomenons of the second mean-value theorem (3.9) if the functions g(u) there
are oscillating functions (remind, only continuity is required) are not yet well illustrated in detail.
We now derive a few general properties of the function u0(0, y) which can be seen in the Figures.
From (4.9) written in the form and by Taylor series expansion according to
Z+∞
0
du Ω(u) cos(uy) = Ω(0)u0(0, y)sin (u0(0, y)y)
u0(0, y)y
=Ω(0)u0(0, y)1−y2
6(u0(0, y))2+. . .,(7.9)
follows from the even symmetry of the left-hand side that u0(0, y) also has to be a function of the
variable ywith even symmetry (notation ∂nu0(0, y)
∂yn≡u(n)
0(0, y))
u0(0, y)=+u0(0,−y),(7.10)
with the consequence
u0(0, y) = u0(0,0) + ∞
X
m=1
u(2m)
0(0,0)
(2m)! y2m, u(2m+1)
0(0,0) = 0.(7.11)
Concretely, we obtain by n-fold differentiation of both sides of (7.9) at y= 0 for the first coefficients
of the Taylor series
Z+∞
0
du Ω(u) = Ω(0)u0(0,0),0 = Ω(0)u(1)
0(0,0),
−Z+∞
0
du Ω(u)u2=Ω(0) u(2)
0(0,0) −1
3(u0(0,0))3,(7.12)
from which follows
u0(0,0) = 1
Ω(0) Z+∞
0
du Ω(u), u(1)
0(0,0) = 0,
u(2)
0(0,0) = −1
Ω(0) Z+∞
0
du Ω(u)u2
| {z }
<0
+1
31
Ω(0) Z+∞
0
du Ω(u)3
| {z }
>0
.(7.13)
26
Since the first sum term on the right-hand side is negative and the second is positive it depends
from their values whether or not u(2)
0(0,0) possesses a positive or negative value. For the special
function Ω(u) in (2.26) which plays a role in the Riemann hypothesis we find approximately
Ω0≡Z+∞
0
du Ω(u)≈0.497121,2!Ω2≡Z+∞
0
du Ω(u)u2≈0.0229719,
Ω(0) ≈1.78679, u0(0,0) ≈0.278220, u(2)
0(0,0) ≈ −0.00567784,(7.14)
meaning that the second coefficient in the expansion of u0(0, y) in a Taylor series in powers of yis
negative that can be seen in the first part of Fig. 7.4. However, as we have seen the proof of the
Riemann hypothesis is by no means critically connected with some numerical values.
In principle, the proof of the Riemann hypothesis is accomplished now and illustrated and we
will stop here. However, for a deeper understanding of the proof it is favorable to consider some
aspects of the proof such as, for example, analogues to other functions with a representation of the
form (3.1) and with zeros only on the imaginary axis and some other approaches although they did
not lead to the full proof that, however, we cannot make here.
8. Equivalent formulations of the main theorems in a summary
In present article we proved the following main result
Theorem 1:
Let Ω(u) be a real-valued function of variable uin the interval 0 ≤u < +∞which is positive
semi-definite in this interval and non-increasing and is rapidly vanishing in infinity, more rapidly
than any exponential function exp (−λu), that means
Ω(u)≥0, Ω(1) (u)≤0,(0 ≤u < +∞),lim
u→+∞
Ω(u)
exp (−λu)= 0,(λ∈R+).(8.1)
Then the following integral with arbitrary complex parameter z=x+ iy
Ξ(z) = Z+∞
0
du Ω(u) ch(uz),(z∈C),(8.2)
is an entire function of zwith possible zeros z=z0=x0+ iy0only on the imaginary axis x= 0
that means
Ξ(z0)=0,⇒z0= iy0,(or x0= 0).(8.3)
Proof:
The proof of this theorem for non-increasing functions Ω(u) takes on Sections 3–5 of this article.
The function Ω(u) in (2.26) satisfies these conditions and thus provides a proof of the Riemann
hypothesis.
Remark:
A analogous theorem is obviously true by substituting in (8.2) ch(u)↔cos(u) and by interchanging
the role of the imaginary and of the real axis y↔x. Furthermore, a similar theorem with a few
peculiarities is true for substituting ch(uz) in (8.2) by sh(uz) .
27
Theorem 1 can be formulated in some equivalent ways which lead to interesting consequences5.
The Mellin transformation ˆ
f(s) of an arbitrary function f(t) together with its inversion is defined
by [32, 33, 34, 35]
ˆ
f(s) = Z+∞
0
dt f (t)ts−1, f(t) = 1
i2πZc+i∞
c−i∞
ds ˆ
f(s)t−s,(8.4)
where the real value chas only to lie in the convergence strip for the definition of ˆ
f(s) by the
integral. Formula (8.2) is an integral transform of the function ch(z) and can be considered as the
application of an integral operator to the function ch(z) which using the Mellin transform ˆ
Ω(s) of
the function Ω(u) can be written in the following convenient form
Ξ(z) = ˆ
Ω∂
∂z zch(z).(8.5)
This is due to
Z+∞
0
du Ω(u) ch(uz) = Z+∞
0
du Ω(u)uz∂
∂z ch(z) = Z+∞
0
du Ω(u)u∂
∂z z−1ch(z),(8.6)
where uz∂
∂z is the operator of multiplication of the argument of an arbitrary function g(z) by the
number u, i.e. it transforms as follows
g(z)→g(uz) = uz∂
∂z g(z),(8.7)
according to the following chain of conclusions starting from the property that all functions zn,(n=
0,1,2, . . .) are eigenfunctions of z∂
∂z to eigenvalue n
z∂
∂z zn=nzn,⇒fz∂
∂z zn=f(n)zn,⇒exp λz ∂
∂z zn=eλzn,⇒
exp λz ∂
∂z g(z) = geλz,⇒uz∂
∂z g(z) = g(uz),(eλ≡u).(8.8)
This chain is almost obvious and does not need more explanations. The operators uz∂
∂z are linear
operators in linear spaces depending on the considered set of numbers u.
Expressed by real variables (x, y) and by ∂
∂z ,∂
∂z∗=1
2∂
∂x −i∂
∂y ,1
2∂
∂x + i ∂
∂y we find
from (8.5)
Ξ(x+ iy) = ˆ
Ω1 + 1
2x∂
∂x +y∂
∂y +i
2y∂
∂x −x∂
∂y ch(x+ iy).(8.9)
From this formula follows that Ξ(iy) may be obtained by transformation of ch(iy) = cos(y) alone
via
Ξ(iy) = ˆ
Ω1 + yi
2∂
∂x −i∂
∂y ch (x+ iy) = ˆ
Ω1+iy∂
∂z ch(z)
=ˆ
Ω∂
∂y ycos(y).(8.10)
5Some of these equivalences now formulated as consequences originate from trials to prove the Riemann hypothesis
in other way.
28
On the right-hand side we have a certain redundance since in analytic functions the information
which is contained in the values of the function on the imaginary axis is fully contained also in
other parts of the function (here of ch(z)).
The most simple transformation of ch(z) is by a delta function δ(u−u0) as function Ω(u) which
stretches only the argument of the Hyperbolic Cosine function ch(z)→ch(u0z). The next simple
transformation is with a function function Ω(u) in form of a step function θ(u0−u) which leads to
the transformation ch(z)→1
zsh(u0z). Our application of the second mean-value theorem reduced
other cases under the suppositions of the theorem to this case, however, with parameter u0=u0(z)
depending on complex variable z.
The great analogy between displacement operators (infinitesimal −i∂
∂x ) of the argument of a
function and multiplication operator (infinitesimal x∂
∂x ) of the argument of a function with respect
to the role of Fourier transformation and of Mellin transformation can be best seen from the
following two relations
Z+∞
−∞
dy f(y)g(x−y) = ˜
f−i∂
∂x g(x),˜
f(t)≡Z+∞
−∞
dy f(y)e−ity,
Z+∞
0
du f (u)g(ux) = ˆ
f∂
∂x xg(x),ˆ
f(s)≡Z+∞
0
du f (u)us−1.(8.11)
We remind you that Mellin and Fourier transform are related by substituting the integration vari-
ables u= eyand the independent variables s=−itand by the substitutions f(ey)↔f(y) and
ˆ
f(−it)↔˜
f(t) in (8.11).
Using the discussed Mellin transformation Theorem 1 can be reformulated as follows
Theorem 1?:
The mapping of the function ch(z) of the complex variable zinto the function Ξ(z) by an operator
ˆ
Ω∂
∂z zaccording to
Ξ(z) = ˆ
Ω∂
∂z zch(z),ˆ
Ω(s)≡Z+∞
0
du Ω(u)us−1,(8.12)
where ˆ
Ω(s) is the Mellin transformation of the function Ω(u) which last possesses the properties
given in Theorem 1 maps the function ch(z) with zeros only on the imaginary axis again into a
function Ξ(z) with zeros only on the imaginary axis.
Proof:
It is proved as a reformulation of the Theorem 1 which is supposed here to be correctly proved.
It was almost evident that the theorem may be formulated for more general functions Ω(u) as
supposed for the application of the second mean-value theorem that was made in Section 6. Under
the suppositions of the theorem the integral on the left-hand side of (8.5) can be transformed by
partial integration to (notation: ˆ
Ω(1)(s)≡R+∞
0du Ω(1) (u)us−1)
Ξ(z) = −1
zZ+∞
0
du Ω(1) (u) sh(uz) = −1
zˆ
Ω(1) ∂
∂z zsh(z).(8.13)
The derivative Ω(1)(u) of the function Ω(u) to the Riemann hypothesis although semi-definite (here
negatively) and rapidly vanishing in infinity is not monotonic and possesses a minimum (see (2.26)
29
and Fig. 2.2). In case of the (modified) Bessel functions we find by partial integration (e.g., [32])
ν!2
zν
Iν(z) = ν!
ν−1
2!1
2!Z+1
0
du 1−u2ν−1
2ch (uz)
=2ν!
ν−3
2!1
2!
1
zZ+1
0
du 1−u2ν−3
2ush (uz),(8.14)
where the functions in the second transform 1−u2ν−3
2ufor ν > 3
2are non-negative but not
monotonic and possess a maximum for a certain value umax within the interval 0 < umax <1. The
forms (8.13) for Ξ(z) and (8.14) suggest that there should be true a similar theorem to the integral
in (8.2) with substitution ch(uz)→sh(uz) and that monotonicity of the corresponding functions
should not be the ultimate requirement for the zeros in such transforms on the imaginary axis.
Another consequence of the Theorem 1 follows from the non-negativity of the squared modulus
of the function Ξ(z) resulting in the obvious inequality
0≤Ξ(z) (Ξ(z))∗
=1
2Z+∞
0
du1Z+∞
0
du2Ω(u1)Ω(u2)
·ch((u1+u2)x) cos((u1−u2)y) + ch((u1−u2)x) cos((u1+u2)y),(8.15)
which can be satisfied with the equality sign only on the imaginary axis z= iyfor discrete values
y=yk(the zeros of Ξ(z=x+ iy)). By transition from Cartesian coordinates (u1, u2) to inertial-
point coordinates (u, ∆u) according to
u=u1+u2
2,∆u=u2−u1, u1=u−∆u
2, u2=u+∆u
2, du1∧du2=du ∧d(∆u),(8.16)
equation (8.15) can be also written
0≤1
2Z+∞
0
du Z+2u
−2u
d(∆u)Ωu−∆u
2Ωu+∆u
2
·ch (2u x) cos (∆u y) + ch (∆u x) cos (2u y).(8.17)
As already said the case of the equality sign in (8.15) or (8.17) can only be obtained for x= 0
and then only for discrete values of y=ykby solution of this inequality with the specialization for
x= 0
0 = Ξ(iy) (Ξ(iy))∗
=1
2Z+∞
0
du1Z+∞
0
du2Ω(u1)Ω(u2)cos((u1−u2)y) + cos((u1+u2)y)
=1
2Z+∞
0
du Z+2u
−2u
d(∆u)Ωu−∆u
2Ωu+∆u
2cos (∆u y) + cos (2u y).(8.18)
A short equivalent formulation of the inequality (8.15) and (8.17) together with (8.18) is the fol-
lowing
30
Theorem 2:
If the function Ω(u) satisfies the suppositions in Theorem 1 then with y−k=−yk
Ξ(x+ iy) (Ξ(x+ iy))∗
>0,⇒∀x6= 0,
x= 0,∀y6=yk,(k=±1,±2, . . .),
= 0,⇒x= 0,∃y=yk,(k=±1,±2, . . .).
(8.19)
Proof:
As a consequence of proved Theorem 1 it is also proved.
The sufficient condition that this inequality is satisfied with the equality sign is that we first set
x= 0 in the expressions on the right-hand side of (8.15) and that we then determine the zeros
y=ykof the obtained equation for Ξ(iy) (Ξ(iy))∗= 0. In case of indefinite Ω(u) there are possible
in addition zeros on the x-axis.
Remark:
Practically, (8.15) is an inequality for which it is difficult to prove in another way that it can be
satisfied with the equality sign only for x= 0. Proved in another way with specialization (2.26) for
Ω(u) it would be an independent proof of the Riemann hypothesis.
Conclusion
We proved in this article the Riemann hypothesis embedded into a more general theorem for a class
of functions Ξ(z) with a representation of the form (3.1) for real-valued functions Ω(u) which are
positive semi-definite and non-increasing in the interval 0 ≤u < +∞and which are vanishing in
infinity more rapidly than any exponential function exp(−λu) with λ > 0. The special Xi function
Ξ(z) to the function Ω(u) given in (2.26) which is essentially the xi function ξ(s) equivalent to the
Riemann zeta function ζ(s) concerning the hypothesis belongs to the described class of functions.
Modified Bessel functions of imaginary argument ’normalized’ to entire functions ( 2
iz)νJν(iz) =
(2
z)νJν(z) for ν≥1
2belong also to this class of functions with a representation of the form (3.1)
with Ω(u) which satisfy the mentioned conditions and in this last case it is well known and proved
in independent way that their zeros lie only on the imaginary axis corresponding to the critical
line in the Riemann hypothesis. Knowing this property of the modified Bessel functions we looked
from beginning for whole classes of functions including the Riemann zeta function which satisfy
analogous conditions as expressed in the Riemann hypothesis. The details of the approach to Bessel
functions and also to certain classes of almost-periodic functions we prepare for another work.
The numerical search for zeros of the Riemann zeta function ζ(s), s =σ+itin the critical strip,
in particular, off the critical line may come now to an end by the proof of the Riemann hypothesis
since its main purpose was, in our opinion, to find a counter-example to the Riemann hypothesis
and thus to disprove it. We did not pay attention in this article to methods of numerical calculation
of the zeros with (ultra-)high precision and for very high values of the imaginary part. However,
the proof if correct may deliver some calculators now from their pain to calculate more and more
zeros of the Riemann zeta function.
We think that some approaches in this article may possess importance also for other problems.
First of all this is the operational approach of the transition from real and imaginary part of
a function on the real or imaginary axis to an analytic function in the whole complex plane.
In principle, this is possible using the Cauchy-Riemann equations but the operational approach
integrates this to two integer instead of differential equations. We think that this is possible also in
31
curved coordinates and is in particular effective starting from curves of constant real or imaginary
part of one of these functions on a curve.
One of the fascinations of prime number theory is the relation of the apparently chaotic dis-
tribution function of prime numbers π(x) on the real axis x≥0 to a fully well-ordered analytic
function, the Riemann zeta function ζ(s), at least, in its representation in sum form as a special
Dirichlet series and thus providing the relations between multiplicative and additive representations
of arithmetic functions.
Acknowledgement
I am very grateful to my son Arne W¨unsche for different help. He helped me very much with the
arrangement of my computer and if there were problems with it and also with the installation and
use of programmes and by often simple questions he gave me new suggestions.
Appendix A.
Transformation of the Xi function
In this Appendix we transform the function ξ(s) defined in (2.8) by means of the zeta function ζ(s)
from the form taken from (2.5) to the form (2.9) using the Poisson summation formula. The Poisson
summation formula is the transformation of a sum over a lattice into a sum over the reciprocal
lattice. More generally, in one-dimensional case the decomposition of a special periodic function
F(q) = F(q+a) with period adefined by the following series over functions f(q+na)
F(q)≡
+∞
X
n=−∞
f(q+na) = F(q+a),(A.1)
can be transformed into the reciprocal lattice providing a Fourier series as follows. For this purpose
we expand F(q) in a Fourier series with Fourier coefficients Fmand make then obvious transfor-
mations (q0+na →q00,
+∞
X
n=−∞ Z(n+1)a
na
dq00 →Z+∞
−∞
dq00 and changing the order of summation and
integration) according to
F(q) =
+∞
X
m=−∞ 1
aZa
0
dq0
+∞
X
n=−∞
f(q0+na) exp −im2πq0
a!
| {z }
=Fm
exp im2πq
a
=1
a
+∞
X
m=−∞ +∞
X
n=−∞ Z(n+1)a
na
dq00 f(q00) exp −im2πq00
a!exp im2πq
a
=1
a
+∞
X
m=−∞
˜
fm2π
aexp im2πq
a,(A.2)
32
where the coefficients Fmof the decomposition of F(q) are given by the Fourier transform ˜
f(k) of
the function f(q) defined in the following way
˜
f(k) = Z+∞
−∞
dxf(q) exp (−ikq),⇒˜
fm2π
a=Z+∞
−∞
dqf (q) exp −im2πq
a.(A.3)
Using the period b=2π
aof the reciprocal lattice relation on the right-hand side of (A.2) it may be
written in the forms
+∞
X
n=−∞
f(q+na) =
+∞
X
n=−∞
fq+n2π
b=b
2π
+∞
X
m=−∞
˜
f(mb) exp (imbq)
=1
a
+∞
X
m=−∞
˜
fm2π
aexp im2π
aq, ab = 2π. (A.4)
In the special case q= 0 one obtains from (A.2) the well-known basic form of the Poisson summation
formula
F(0) ≡
+∞
X
n=−∞
f(na) = b
2π
+∞
X
m=−∞
˜
f(mb), ab = 2π. (A.5)
Formula (A.5) applied to the sum
+∞
X
n=−∞
exp −πn2q2corresponding to f(q) = exp −πq2
with Fourier transform ˜
f(k) = exp −k2
4πprovides a relation which can be written in the following
symmetric form (we need it in the following only for q≥0)
Ψ(q)≡p|q|(1
2+∞
X
n=1
exp −πn2q2)=p|q|
2
+∞
X
n=−∞
exp −πn2q2
=1
2p|q|
+∞
X
m=−∞
exp −πm2
q2=1
p|q|(1
2+∞
X
m=1
exp −πm2
q2)≡Ψ1
q.(A.6)
This is essentially a transformation of the Theta function ϑ3(u, q) in special case 2Ψ(q)
√|q|≡ϑ30,e−πq2.
We now apply this to a transformation of the function ξ(s).
From (2.9) and (2.5) follows
ξ(s) = s(s−1) Z+∞
0
dq qs−1∞
X
n=1
exp −πn2q2
=s(s−1) (Z1
0
dq qs−1∞
X
n=1
exp −πn2q2+Z+∞
1
dq qs−1∞
X
n=1
exp −πn2q2).(A.7)
The second term in braces is convergent for arbitrary qdue to the rapid vanishing of the summands
of the sum for q→ ∞. To the first term in braces we apply the Poisson summation formula (A.5)
33
and obtain from the special result (A.6)
Z1
0
dq qs−1∞
X
n=1
exp −πn2q2=Z1
0
dq qs−1(1
21
q−1+1
q
∞
X
m=1
exp −πm2
q2)
=1
2s(s−1) +Z+∞
1
dq0q0−s∞
X
m=1
exp −πm2q02,(A.8)
with the substitution q=1
q0of the integration variable made in last line. Thus from (A.7) we find
ξ(s) = 1
2−s(1 −s)Z+∞
1
dq qs+q1−s
y
∞
X
n=1
exp −πn2q2.(A.9)
With the substitution of the integration variable
q= eu,(q≥0,⇔ −∞ < u < +∞),(A.10)
and with displacement of the complex variable sto z≡s+1
2and introduction of Ξ(z) instead of
ξ(s) this leads to the representation
Ξ(z)≡ξ1
2+z=1
2−21
4−z2Z+∞
0
du ch (uz) e u
2
∞
X
n=1
exp −πn2e2u,(A.11)
given in (2.24). In Appendix B we transform this representation by means of partial integration
to a form which due to symmetries is particularly appropriate for the further considerations about
the Riemann zeta function.
Using the substitution (A.10) we define a function Φ(u) by means of the function Ψ(y) in (A.6)
as follows
Φ(u)≡Ψ(eu),⇒Φ(0) = Ψ(1) = 0.543217,(A.12)
and explicitly due to Poisson summation formula
Φ(u)≡eu
2(1
2+∞
X
n=1
exp −πn2e2u)=1
2eu
2
+∞
X
n=−∞
exp −πn2e2u
=1
2e−u
2
+∞
X
n=−∞
exp −πn2e−2u= e−u
2(1
2+∞
X
n=1
exp −πn2e−2u).(A.13)
From Ψ(y) = Ψ1
yaccording to (A.6) follows that Φ(u) is a symmetric function
Φ(u) = Φ(−u) = Φ(|u|).(A.14)
Therefore, all even derivatives of Φ(u) are also symmetric functions, whereas all odd derivatives of
Φ(u) are antisymmetric functions (we denote these derivatives by Φ(n)(u)≡∂nΦ
∂u (u))
Φ(2m)(u)=+Φ(2m)(−u) = Φ(2m)(|u|),
Φ(2m+1) (u) = −Φ(2m+1) (−u),⇒Φ(2m+1) (0) = 0,(m= 0,1,2, . . .).(A.15)
34
Explicitly, one obtains for the first two derivatives
Φ(1) (u) = 1
2eu
2(1
2+∞
X
n=1 1−4πn2e2uexp −πn2e2u),
Φ(2) (u) = 1
4eu
2(1
2+∞
X
n=1 1−64πn2e2u+4πn2e2u2exp −πn2e2u).(A.16)
As a subsidiary result we obtain from vanishing of the odd derivatives of Φ(u) at u= 0 that means
from Φ(2m+1) (0) = 0,(m= 0,1,2, . . .) an infinite sequence of special sum evaluations from which
the first three are
∞
X
n=1 4πn2−1exp −πn2=1
2,
∞
X
n=1 4πn23−15 4πn22+ 31 4πn2−1exp −πn2=1
2.(A.17)
We checked relations (A.17) numerically by computer up to a sufficiently high precision. We also
could not find (??) among the known transformations of theta functions. The interesting feature
of these sum evaluations is that herein power functions as well as exponential functions containing
the transcendental number πin the exponent are involved in a way which finally leads to a rational
number that should also be attractive for recreation mathematics. In contrast, in the well-known
series for the trigonometric functions one obtains for certain rational multiples of πas argument
also rational numbers but one has involved there only power functions with rational coefficients
that means rational functions although an infinite number of them.
Using the function Φ(u) the function Ξ(z) in (A.11) can be represented as
Ξ(z) = 1
2+ 2 z2−1
4Z+∞
0
du Φ(u)−1
2eu
2ch (uz)
=1
2+ 2 Z+∞
0
du Φ(u)−1
2eu
2 ∂2
∂u2−1
4ch (uz).(A.18)
From this we obtain by partial integration
Ξ(z)=2Z+∞
0
du ∂2
∂u2−1
4Φ(u)−1
2eu
2ch (uz),(A.19)
where the contribution from the lower integration limit at u= 0 has exactly canceled the constant
term 1
2on the right-hand side of (A.18) and the contributions from the upper limit x→+∞is
vanishing. Using (A.16) we find with abbreviation Ω(u) according to
Ω(u)≡2∂2
∂u2−1
4Φ(u)−1
2eu
2,(A.20)
the following basic structural form of the Xi function
Ξ(z) = Z+∞
0
du Ω(u) ch (uz),(A.21)
35
with the following explicit representation of Ω(u)
Ω(u)=2Φ(2) (u)−1
2Φ(u)
= 4eu
2
∞
X
n=1
πn2e2u2πn2e2u−3exp −πn2e2u>0.(A.22)
Since according to (A.15) the even derivatives of Φ(u) are symmetric functions it follows from
relation (A.22) that Ω(u) is also a symmetric function and (2.27) holds. This is not immediately seen
from the explicit representation (A.22). Furthermore, Ω(u) is positively definite for −∞ < u < +∞
since the factor 2πn2e2u−3in (A.22) is positive for n≥1 and u≥0 and all other factors too.
It goes rapidly to zero for u→ ±∞, more rapidly than any exponential function exp (−γ|x|ρ) with
arbitrary γ > 0 and arbitrary ρ > 0 due to factors exp −πn2e2uin the sum terms in (A.22). For
the first derivative of Ω(u) we find
Ω(1)(u)=2Φ(3) (u)−1
2Φ(1) (u)
=−2eu
2
∞
X
n=1
πn2e2u8πn2e2u2−30πn2e2u+ 15exp −πn2e2u
=−Ω(1)(−u),⇒Ω(1)(0) = 0, Ω(1)(u)<0,(u > 0) .(A.23)
It is vanishing for u= 0 due to its antisymmetry and negatively definite for u > 0 as the negative
sign of Ω(2)(0) together with considerations of the sum for u > 0 show (i.e., the polynomial
f(N)≡8π2N2−30πN + 15 ≥0 for N≥15+√105
8π≈1.00454 and negativity is already obtained
taking the first two sum terms to n= 1 and n= 2 alone). Thus Ω(u) is monotonically decreasing
for u≥0. A few approximate numerical values of parameters for the function Ω(u) are
Ω0≡Z+∞
0
du Ω(u) = −Z+∞
0
du Ω(1) (u)u= 0.497121,
Ω(0) = −Z+∞
0
dx Ω(1) (u) = 1.78679, Ω(1)(0) = −Z+∞
0
du Ω(2) (u) = 0.(A.24)
In next Appendix we consider the transition from analytic functions given on the real or imag-
inary axis to the whole complex plane.
Appendix B.
Transition from analytic functions on real or imaginary axis to
whole complex plane
The operator ∂
∂x is the infinitesimal displace operator and exp−x0∂
∂x the finite displacement
operator for the displacement of the argument of a function f(x)→f(x−x0). In complex analysis
the real variable xcan be displaced with sense for an analytic function to the complex variable
z=x+ iyin the whole complex plane by
exp iy∂
∂x xexp −iy∂
∂x =x+iy
1! ∂
∂x , x+(iy)2
2! ∂
∂x ∂
∂x , x+. . . =x+ iy, (B.1)
36
where [A, B]≡AB −BA denotes the commutator of two operators Aand B, in particular ∂
∂x , x=
1 and (B.1) may be written in the form
exp iy∂
∂x x= (x+ iy) exp iy∂
∂x .(B.2)
Analogously, the transition from the variable yon the imaginary axis iyto the variable z=
i(y−ix) = x+ iyin the whole complex plane may be written as
exp −ix∂
∂y iy= (x+ iy) exp −ix∂
∂y ,(B.3)
In the following we consider only the case (B.2) since the case (B.3) is completely analogous with
simple substitutions.
We wrote the equations (B.1), (B.2) and (B.3) in a form which we call operational form and
meaning that they may be applied to further functions on the left-hand and correspondingly right-
hand side6. It is now easy to see that an analytic function w(z) = w(x+ iy); ∂
∂z∗w(z) = 0 can be
generated from w(x) on the x-axis in operational form by
exp iy∂
∂x w(x) = w(x+ iy) exp iy∂
∂x ,(B.4)
and analogously from w(iy) on the imaginary axis by
exp −ix∂
∂x w(iy) = w(x+ iy) exp −ix∂
∂y .(B.5)
Writing the function w(z) with real part u(x, y) and imaginary part v(x, y) in the form
w(x+ iy) = u(x, y)+iv(x, y),(w(x+ iy))∗=u(x, y)−iv(x, y),⇒
u(x, y) = 1
2w(x+ iy)+(w(x+ iy))∗, v(x, y) = −i
2w(x+ iy)−(w(x+ iy))∗,(B.6)
we find from (B.4)
exp iy∂
∂x w(x) = exp iy∂
∂x u(x, 0) + iv(x, 0)
=u(x+ iy, 0) + iv(x+ iy, 0)exp iy∂
∂x
=u(x, y)+iv(x, y)exp iy∂
∂x ,(B.7)
and correspondingly
exp iy∂
∂x w(x)∗= exp −iy∂
∂x u(x, 0) −iv(x, 0)
=u(x−iy, 0) −iv(x−iy, 0)exp −iy∂
∂x
=u(x, y)−iv(x, y)exp −iy∂
∂x .(B.8)
6Non-operational form would be if we write, for example, exp iy∂
∂x x=x+ iyinstead of (B.2) which is correct
but cannot be applied to further functions f(x)6=const ·1, for example to f(x) = x.)
37
From (B.7) and (B.8) follows forming the sum and the difference
cos y∂
∂x u(x, 0) −sin y∂
∂x v(x, 0) = u(x, y) cos y∂
∂x −v(x, y) sin y∂
∂x ,
sin y∂
∂x u(x, 0) + cos y∂
∂x v(x, 0) = u(x, y) sin y∂
∂x +v(x, y) cos y∂
∂x .(B.9)
These are yet operational identities which can be applied to arbitrary functions f(x). Applied to
the function f(x) = 1 follows
cos y∂
∂x u(x, 0) −sin y∂
∂x v(x, 0) = u(x, y),
sin y∂
∂x u(x, 0) + cos y∂
∂x v(x, 0) = v(x, y).(B.10)
In full analogy we may derive the continuation of an analytic function from the imaginary axes
z= iyto the whole complex plane z=x+ iyin operational form
cos x∂
∂y u(0, y) + sin x∂
∂y v(0, y) = u(x, y) cos x∂
∂y +v(x, y) sin x∂
∂y ,
−sin x∂
∂y u(0, y) + cos x∂
∂y v(0, y) = −u(x, y) sin x∂
∂y +v(x, y) cos x∂
∂y ,(B.11)
and this applied to the function f(y)=1
cos x∂
∂y u(0, y) + sin x∂
∂y v(0, y) = u(x, y),
−sin x∂
∂y u(0, y) + cos x∂
∂y v(0, y) = v(x, y).(B.12)
It is easy to check that both (B.10) and (B.12) satisfy the Cauchy-Riemann equations
∂u
∂x (x, y) = ∂ v
∂y (x, y),∂u
∂y (x, y) = −∂v
∂x (x, y),(B.13)
and it is even possible to derive these relations from these equations by Taylor series expansions of
u(x, y) and v(x, y) in powers of yor xin dependence from which axis we make the continuation to
the whole complex plane. For example, in expansion in powers of ywe obtain using (B.13) (and
the resulting equations ∂2
∂x2+∂2
∂y2u(x, y) = ∂2
∂x2+∂2
∂y2v(x, y) = 0 from them)
w(x+ iy) = ∞
X
m=0
y2m
(2m)! ∂2m
∂y2mu(x, y)y=0
+∞
X
m=0
y2m+1
(2m+ 1)! ∂2m+1
∂y2m+1 u(x, y)y=0
+i (∞
X
m=0
y2m
(2m)! ∂2m
∂y2mv(x, y)y=0
+∞
X
m=0
y2m+1
(2m+ 1)! ∂2m+1
∂y2m+1 v(x, y)y=0),(B.14)
that can be written in compact form
w(x+ iy) = cos y∂
∂x u(x, 0) −sin y∂
∂x v(x, 0)
+i cos y∂
∂x v(x, 0) + sin y∂
∂x u(x, 0)=u(x, y)+iv(x, y),(B.15)
38
and is equivalent to (B.10). Analogously by expansion in powers of xas intermediate step we obtain
w(x+ iy) = cos x∂
∂y u(0, y) + sin x∂
∂y v(0, y)
+i cos x∂
∂y v(0, y)−sin x∂
∂y u(0, y)=u(x, y)+iv(x, y),(B.16)
that is equivalent to (B.11). Therefore, relations (B.15) and (B.16) represent some integral forms
of the Cauchy-Riemann equations.
In cases if one of the functions u(x, 0) or v(x, 0) in (B.10) or u(0, y) or v(0, y) in (B.12) is
vanishing these formulae simplify and the case v(0, y) = 0 is applied in Sections 4–6. We did not
find up to now such representations in textbooks to complex analysis but it seems possible that
they are somewhere.
39
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41
