Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
“Riemann’s Hypothesis, a Second_Three_Pages_Proof”
by Fausto Galetto,
independent researcher, past professor of Quality Management at Politecnico of Turin
fausto.galetto@gmail.com
Abstract
After the very short and simple first proof of the RC (short for “Riemann Conjecture”), using the
series expansion about z=1/2 of the Xi function (z) and the Hilbert Spaces Theory, now the author
provides a new, second, proof, using the same series expansion about z=1/2 of the Xi function (z)
and Algebra. This new proof is so simple that the author wonders why a great mathematician like
Riemann did not see it; therefore F. Galetto thinks that somewhere in the purported proof there
should be an error.
1. Introduction
This paper is a preprint of a future paper…
On April 14th 2019 the author provided a (first) “one page proof [5] based on the fact
(found in a Wolfram MathWorld page [4]) that the entire function (z) can be expressed as
x(z)=∑(− 0.5)
a “series expansion about z=1/2” with suitable coefficients an,
shown later. Working on this fact the author showed a very simple (first) proof of the
Riemann Conjecture.
To get our first proof in [5] the author modified the expansion from x(z)=∑(−
0.5) to x(z)=∑(− 0.5) /(2)!
, where (− 0.5) /(2)!were the entries of the
expansion of the analytic function cosh(− 0.5), so that he derived the relationship
x(z)= 4 ∫[/()]
cosh[(− 0.5)0.5 ln()] . . Then he considered the
two infinite dimensional vectors (the 1st real and the 2nd complex) b=[b0, b2, …, b2n,…, …,
b∞] and c(z)=[1, (z-0.5)2/2!, (z-0.5)4/4!,…, (z-0.5)2n/(2n)!,….], such that (z) was the inner
product [or scalar product] c(z)b=(z) with norms ||b||<∞ and ||c(z)||<∞, because
()( )<cosh[(x-0.5)2+y2]<∞. Letting z1=x1+y and z2=x2+y be two zeros symmetric to the
Critical Line, (z1)=(z2)=0, that is ()=(1 −
)= 0, which means orthogonality in
an l2Hilbert space (on the field C of complex numbers), the two vectors ()and (1 −
)were both orthogonal to b so proving that RH is true.
Now, using the same “series expansion about z=1/2” x(z)=∑(− 0.5)
, the
author shows a new second proof of the Riemann Hypothesis (Conjecture).
It is well known that for over a century mathematicians have been trying to prove the
Riemann Conjecture (also known as Hypothesis), RC for short, a conjecture claimed by
Riemann [professor at University of Gottingen in Germany], near 1859 in a 8-page paper
“On the number of primes less than a given magnitude” shown at Berlin Academy, and
dated/published in 1859; at that time, Bernhard Riemann conjectured that all the zeros of
the zeta function should have their real part equal to 0.5; unfortunately he left the
problem aside by writing “... it is very probable that all roots are real. Without doubt it would
be desirable to have a rigorous proof of this proposition; however I have left this research
aside for the time being after some quick unsuccessful attempts, because it appears to be
unnecessary for the immediate goal of my study...”. The comment was related to the real
function (t) [named (t) by B. Riemann] obtained from the zeta function (1/2+it) with
=0.5, (the line =Re(z)=1/2 is named Critical Line).
Figure 1 From the original B. Riemann manuscript
2. The proof of RC
To get our result we use the entire function (z) which has the functional equation
(z)= (1 − z)and is analytic, defined by (z)=z(z − 1)Γ (z)[no poles].
The nontrivial zeros of the Riemann Zeta Function (z) exactly correspond to those of
(z); putting z=1/2+it (i.e. for real t, the z points are on the Critical Line) the roots of
(1/2+it) are the same as those of (1/2+it); moreover (1/2+it)=(t) is a purely real
function, with (see fig. 1, with different notation) Ξ(t)= − ( +)Γ −
− it .
The coefficients an, in Wolfram MathWorld [4], are given by the formula, which
depends only on 2n, = 4 ∫ [ / ()]
[ . ()]
( )! . ; once computed a2n
we have only to compute (z).
To get our second proof we write explicitly some items of the expansion
x
(1)
as follows
()= + − + − + − + ⋯ (1’)
To go on we assume that RH is false: there are at least four points (zeros), out of the
Critical Line, CL; at these points the Xi function is zero. They are shown in figure 2, where
the Critical Line, CL, intercepts the real axe in the point P(0.5, 0), the red dot.
The 4 points are
Symmetric to the Critical Line
Symmetric to the real axe
Symmetric to the point P(0.5, 0), interception of the Critical Line with the real
axe
The 4 points on the s plane are the black dots s,s
[complex conjugate of s], 1 − s,
1 − s
[complex conjugate of 1 − s].
In figure 2 it is “only indicated” the function (), near the black dots; the function
cannot be shown because we would need a 4-dimensional space [Re(s), Im(s), Re (),
Im ()].
If we set a=(s-0.5) and b=(s
-0.5), we have (s)= 0 and (s
)= 0; therefore
(2)
Any power difference can be splitted into the product
(3)
IF n is odd the splitting provides the product
(4)
with P1(a, b) and P2(a, b) suitable polynomials.
IF, on the contrary, n is even and a suitable power of 2, say 2k, then we can continue
the splitting at its end
(5)
with P3(a, b) and P4(a, b) … suitable polynomials.
IF none of the previous two cases happens, sooner or later, there will be an exponent
odd, say m, for which we can apply (4).
The end of the story is that we have, for any couple of points a and b
x x (6)
We use for any suitable couple of zeros…
Figure 2 Four points satisfying (s)=0
Now we use (6) for various couples of zeros, as depicted in figure 2.
Let’s consider first the two zeros on the left of the CL
=+ and
=− (7a)
By our hypothesis we have
()= 0 and (
)= 0 (8a)
and therefore (here we provide any transformation! We will not do later)
0 = ()−(
)=[− 0.5 − (
− 0.5)]{− 0.5 +
−
0.5}Φ(,) = [−
]{+
− 1}Φ(,) = [2 ]{2− 1}Φ(,)
(9a)
from which we get
(10a)
Let’s consider secondly the two zeros on the right of the CL
1 − = 1 − − and 1 −
= 1 − + (7b)
By our hypothesis we have
(1 − )= 0 and (1 −
)= 0 (8b)
and therefore (here we do not provide any transformation, as said before…)
0 = (1 − )−(1 −
)=()−(
)=[1 − − 0.5 − (1 −
− 0.5)]{1 −
− 0.5 + 1 −
− 0.5}Φ(,)=[−+
]{−−
+ 1}Φ(,)=
[−2 ]{−2+ 1}Φ(,)(9b)
Re()
Im(
)
Points of the Xi function
ξz =
(z−1)Γ 2+ 1 z
ξ
s
ξ
1
−
s
ξ
s
ξ
1
−
s
ξ z = ξ 1 − z
P(0.5, 0)
from which we get
(10b)
Let’s consider thirdly the two zeros above the real axe
=+ and 1 −
= 1 − + (7c)
By our hypothesis we have
()= 0 and (1 −
)= 0 (8c)
and therefore (here we do not provide any transformation, as said before…)
0 = ()−(1 −
)=[− 0.5 − (1 −
− 0.5)]{− 0.5 + 1 −
−
0.5}Φ(,)=[− 1 +
]{−
}Φ(,)=[2− 1]{2 }Φ(,)(9c)
from which we get
(10c)
Let’s consider fourthly the two zeros below the real axe
=− and 1 − = 1 − − (7d)
By our hypothesis we have
(
)= 0 and (1 − )= 0 (8d)
and therefore (here we do not provide any transformation, as said before…)
0 = (
)−(1 − )=()−(1 −
)=[
− 0.5 − (1 − − 0.5)]{
−
0.5 + 1 − − 0.5}Φ(,)=[
−1+]{
−}Φ(,)=[2−
1]{−2 }Φ(,)(9d)
from which we get
(10d)
There no need to verify the other two couples due to the functional relationship
We can repeat the same arguments for any zero of the function (): the 4 assumed
zeros are actually “two” zeros (complex conjugated) on the Critical Line: x=1/2, which
contradicts our previous hypothesis that RH was false . Then the Riemann’s Hypothesis
(Riemann’s Conjecture) is TRUE, because all the zeros of the function ()are on the
Critical Line.
3. Conclusion
We assumed that RC was false (see figure 2). Using the series expansion of ()
about s=0.5 [formula 1] we found that actually =0.5 for any zero sm,(s)= 0, out of
the Critical Line: x=1/2. Since it was proved (Titchmarsh 1986) that there are infinite zeros
of the Riemann zeta function (s) in the Critical Strip, there are infinite values sk=k+ik
such that (sk)=0=(sk), [0 < k< 1]. For any ksuch that (sk)=(sk)=0, either there is only
one zero with k=1/2 (on the Critical Line) or four zeros, two couples symmetric to the
critical line and two couples symmetric to the real axe, with different real parts kand 1-k,
that we proved, above, impossible for a particular value ksuch that (sk)=0=(sk). Since
we can repeat the same argument for any quadruple of zeros assumed symmetric [for any
ikand -ik, such that (sk)=0, for any nontrivial zero sk] to the Critical Line and to the real
axe. any zero has =1/2 and therefore RH is true.
References
1. Titchmarsh E.C., The Theory of the Riemann Zeta-Function, CLARENDON PRESS,
OXFORD, 1986
2. Fausto Galetto, Riemann Hypothesis proved. Academia Arena 2014;6(12):19-22].
(ISSN 1553-992X), http://www.sciencepub.net/academia,fausto.galetto@gmail.com
3. Fausto Galetto, A new proof of the Riemann Hypothesis, Research Trends on
Mathematics and Statistics Vol. 3, 23-35, 2019 and HAL archive, 2018
4. Wolfram MathWorld, Xi function, 2019
5. Fausto Galetto, Riemann’s Conjecture, a “One Page Proof (new)”, HAL archive <>,
2019 and Academia.edu 2019
6. Luigi Amerio, Analisi Matematica, vol. III, parte 1°, UTET, 1989