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A New Proof of H.Hasse’s Global Expression for the
Riemann’s Zeta Function .
Demetrios P. Kanoussis
Kalamos Attikis, Greece
e-mail: dkanoussis@gmail.com
Abstract
In this paper we present a new proof of Hasse’s global representation for the Riemann’s
Zeta function , , originally derived in 1930 by the German mathematician
Helmut Hasse. The key idea in our analysis is that the Hurwitz zeta function ,
originally defined as
for and real , can be obtained as
a solution of the difference equation .
This solution provides the analytic continuation of for all complex .
The Riemann’s zeta function is then obtained as . At negative integers the
Riemann’s zeta function reduces to a finite sum whose value is given in terms of the
Bernoulli numbers.
Keywords and Phrases: Hurwitz zeta function, Riemann zeta function, Bernoulli numbers,
Difference equations, completely monotonic functions, Gamma function.
2010 Mathematics Subject Classification: 39A60, 11M35
1. Introduction
Let be a real valued function defined on the interval . Three classical
operators which can apply to , are the derivative operator
the shift operator
(1-1)
and the difference operator
(1-2)
where is the identity operator.
Higher order operators are defined similarly,
(1-3)
, and (1-4)
, (1-5)
where is any positive integer.
For completeness we define, (for ),
, , and .
If and are any two positive integers, it is easily shown that the operators and
commute, i.e.
, (1-6)
provided that is times differentiable on .
In some recent papers ( [10], [11] and [12] ) the inverse Taylor operator was defined as
(1-7)
and it was further shown that for a wide class of functions, this operator is equivalent to the
derivative operator .
Definition 1. We say that a real valued function , defined on , belongs to the class
, if
, (1-8)
for all .
In particular represents the class of functions for which the expansion (1-8) is valid
for all real values of , , while represents the class of functions satisfying
(1-8) for all positive values of , .
This paper relies heavily on the work developed in [10] where it was shown that
1) All polynomials are in ,
2) The exponential function ,
3) The trigonometric functions ) and are in ,
4) The function and all its antiderivatives are in ,
5) If is any real number , the function ,
6) If , , is the Laplace Transform of a positive function , , then
and any antiderivative of , are in , ().
In the sequel of this paper we shall refer often to Theorems and/or Equations quoted in [10]
by their respective numbers, (Theorem number and/or Equation number).
Also for brevity and notation economy reasons, in the rest of this paper, we shall make use
of the simplified notation or and or , instead of and
respectively.
2. Further properties of the function .
By Theorem 14 in [10], the function , meaning that for and ,
(2-1)
Another series closely related to the series in (2-1), is
(2-2)
Since this series will play an important role in our subsequent analysis, we proceed by
proving the following
Theorem 1. (a) If is any non-negative integer , the series in (2-2)
terminates and represents a polynomial in , of degree .
(b) For all other real values of , the series in (2-2) is absolutely and uniformly convergent on
.
Proof: (a) Let be any non-negative integer. Then , for , i.e.
for , therefore the series in (2-2) reduces to the finite sum
, which obviously represents a polynomial in , of degree .
(b) 1) Let us now assume that . We may set , where . The function
is completely monotonic on and at the same time , by
Theorem 5 in [10]. By virtue of Theorem 7 in [10], the infinite series
converges absolutely and uniformly on .
We note that for and
,
(2-3)
by virtue of Theorem 6 in [10], and since the series
converges to
(since ), application of the Weierstrass M- Test shows that the series in
(2-2) is absolutely and uniformly convergent, and this completes the proof of Theorem 1,
for .
We now assume that is any non-integer positive number.
2) We first consider the case where or equivalently, .
The function , (Theorem 5 in [10]) and at the same time is
completely monotonic on . The function is one first antiderivative of
, and by virtue of Theorem 9 in [10],the series
is absolutely and uniformly convergent on .
We note that for and
. (2-4)
In deriving (2-4) we have made use of the fact that , (Theorem 8 in
[10] applied for ) and equation (3-10) in [10].
The series
converges to , (since ), and therefore
application of the Weierstrass M-Test implies that the series
converges absolutely and uniformly on , and evidently so does the
series in (2-2), and this completes the proof of Theorem, for .
3) We next assume that , or equivalently . The function
, (Theorem 5 in [10]), and is completely monotonic on .
The function is one second antiderivative of , and by virtue of Theorem 9 in
[10], the series
is absolutely and uniformly convergent on .
We note that for and
. (2-5)
In deriving (2-5) we have made use of the fact that , (Theorem 8 in
[10] applied for ) and equation (3-13) in [10]. The series
converges to ,
since .
Application of the Weierstrass M-Test, again, shows that the series
converges absolutely and uniformly on , and evidently so does the
series in (2-2), and this completes the Proof of Theorem 1, for .
4) Proceeding similarly, step by step, one may prove Theorem 1, for , then for
, …,etc, completing thus the proof of Theorem 1 for all real .
Theorem 2. If is any fixed positive number, then
(2-6)
Proof: Since , (from Theorem 6 in [10] ), equation (2-3) implies,
and adding term wise from up to , we obtain,
, (2-7)
since .
For fixed positive, as , and , and taking (2-7) into
account, (2-6) follows immediately.
3 . The function , ,
In our so far analysis we have dealt with the function with and real.
One obvious generalization, useful for our subsequent analysis, is to consider complex, i.e
to consider the function
, , . (3-1)
The following Theorem is a generalization of Theorem 14, in [10], from real to complex.
Theorem 3. If is any complex number, .
Proof: Let be any complex number, with , (since corresponds to
real, and , by Theorem 14 in [10]).
1) If , then (see [17]),
(3-2)
where is the well known Gamma function, and setting , we
have,
, , , (3-3)
from which,
(3-4)
Let us now consider the infinite series,
. (3-5)
If we call ,
, ( for brevity),
it is easily shown, (taking (1-17) in [10] into account) that
therefore by the Fubini-Tonelli Theorem, we may interchange summation with integration
in (3-5), to obtain,
(again by virtue of (1-17) in [10]), or
and this proves that , for the case where .
We now proceed to prove Theorem 3, for the case where .
The proof is accomplished step by step, (vertical strip by vertical strip), as shown in the
sequel.
2) Let , with .
We may set,
, therefore , with .
From (1-15) in [10],
, and (3-6)
which simplifies to
(3-7)
As , , as proved in Appendix 1, while
since for the function , as proved in part (1).
Therefore as , (3-7) implies
meaning that for , .
3) We next consider the case , with We may set,
, therefore , with .
Working as in part (2), equation (3-6) implies,
(3-8)
As , , as proved in Appendix 1, while
since for , , as proved in part (2).
Therefore as , (3-8) implies
meaning that for , .
4) Proceeding similarly, step by step, the validity of Theorem 3 is thus established for all
complex values of , (with ).
Next Theorem is a generalization of Theorem 1, from Real to Complex.
Theorem 4. For any and , with , the series
(3-9)
is absolutely convergent, (and therefore convergent).
Proof: 1) We assume that , since the case , corresponding to
real, is covered in Theorem 1.
2) The difference of , can be expressed as (see [9])
(3-10)
and similarly, for , with ,
(3-11)
(We note that if the corresponding series in (2-2) ,with
(real), terminates and represents a polynomial in of degree ).
From (3-10) and (3-11), we have,
(3-12)
where
(3-13)
For a fixed , and in the neighborhood of ,
and
,
therefore from (3-13) we have,
(3-14)
Since the sequence is convergent to the positive number
, as shown in (3-14), it
will be absolutely bounded, i.e. there exists a positive number , independent of , such
that
. (3-15)
In view of (3-15), equation (3-12) implies
(3-16)
and since the series
is absolutely convergent, by virtue of Theorem 1,
the series in (3-9) is absolutely convergent for any and , with
and .
Finally we prove Theorem 4, for the case and .
3) Let us first consider the case where i.e. . From (3-10) we
have,
(3-17)
Also application of (3-11) with , yields,
(3-18)
From (3-17), (3-18) and the fact that
, for
one easily obtains that for
(3-19)
where
(3-20)
and therefore,
. (3-21)
This shows that the sequence is absolutely bounded, i.e. there exists a positive constant
, independent of , such that
. (3-22)
In view of (3-22), equation (3-19) implies that for ,
and since the series
is absolutely convergent, by Theorem 1, the
series in (3-9), for any and , is absolutely convergent as well.
4) We next consider the case where . From (3-10) we have,
,
(3-23)
while from (3-11), with , we have,
. (3-24)
Since for ,
one easily obtains from (3-23) and (3-24) that for , we have
(3-25)
where
, (3-26)
and therefore,
(3-27)
Reasoning as in parts (2) and (3), one easily shows that the series in (3-9), for any and
, is absolutely convergent.
5) Proceeding similarly, one can prove Theorem 4, for , and
this finally completes the proof of Theorem 4, for any complex number .
4. The Hurwitz zeta function , .
The Hurwitz zeta function , one of the most fundamental transcendental functions
in mathematics, was originally defined for and real , by the convergent series,
(4-1)
The study of has attracted the attention of many researchers, for a wide variety of
problems, related either to the analytic continuation of with respect to , to the
whole complex plane, (except ), or to the evaluation of certain types of definite
integrals and infinite sums, (see [3], [5], [6], [7 ] and [15]).
In this paper, a particularly simple method is developed, by means of which one may
naturally extend , with respect to , to the whole complex plane, except for a simple
pole at , with residue .
The scheme in our approach will be, to extent firstly , as defined in (4-1), from real
to all , and subsequently to all .
Theorem 5. For any and real ,
(4-2)
Proof: Let us consider the function and real . Obviously,
is positive, decreasing and concave upwards. If is any positive number, and ,
we have
and adding term wise from up to , we obtain
(4-3)
As , (4-3) implies
and the proof is completed.
Theorem 6. (i) For a fixed , the
(ii) For a fixed , the ,
(iii) If , the .
Proof: The proof of (i), (ii), (iii) follows easily, as an immediate consequence of (4-2).
Next Theorem is of fundamental importance, for our subsequent analysis.
Theorem 7. The Hurwitz zeta function as defined in (4-1), for and real ,
can be expressed equivalently as
, (4-4)
where the operator operates on the variable.
Proof: The key idea in proving (4-4) is to note that the function in (4-1) satisfies the
difference equation,
, , or
. (4-5)
However the function , by virtue of Theorem 14, in [10], meaning that
and hence (4-5) implies,
(4-6)
since the series inside the braces converges, by Theorem 1, and therefore,
, (4-7)
where up to this point, is an arbitrary constant.
In order to determine , we consider the limiting case in (4-7), as . For a fixed ,
the , by virtue of Theorem 6(i), and also
by virtue of Theorem 2, with , therefore , and finally,
, (4-8)
and this completes the proof of Theorem 7.
However the series in (4-4), (which coincides with as defined in (4-1) for all real
), by virtue of Theorem 1, converges for all real values of , ( due to the
term
in (4-4) ), and still the same series converges for all complex values of ,
by virtue of Theorem 4.
This fact enables one to extend the definition of with respect to , from real to
any complex number .
We are thus led to the following Theorem,
Theorem 8. (i) The Hurwitz zeta function with and , can be
expressed as
(4-9)
(ii) For and real , the in (4-9) coincides with as originally defined in
equation (4-1).
Proof: The proof follows easily from Theorems 4 and 7.
Next Theorem reveals the intrinsic connection between the Hurwitz zeta function , as
defined in (4-9), at negative integers of the argument , and the Bernoulli Polynomials.
Theorem 9. If is the Bernoulli polynomial of degree , then
(4-10)
Proof: It is known that the Bernoulli polynomials satisfy the following identities ([1])
, and (4-11)
, (4-12)
For , the series in (4-9) terminates, the last term being
, and
or taking (4-11) into account,
, (4-13)
and since the polynomial , by Theorem 2, in [10],we have from (4-13) and
(4-12),
and the proof is completed.
5. The Riemann’s zeta function .
The mathematical literature concerning the Riemann’s zeta function is enormous. The
original definition and many deep properties of , are traced back in the work of Leonard
Euler, who defined by means of the convergent infinite series,
, (5-1)
where is a positive integer , i.e. .
Dirichlet seems to be the first to consider for all positive real numbers , (the series in
(5-1) obviously converges for any real ), while Riemann made the crucial step towards
considering for complex values of the argument .
Much of the recent mathematical research is directed towards extending , from real
, to any , ( [2], [8], [13], [16]), not to mention the still open problem about
the position of the non trivial roots of the Riemann’s zeta function, (Riemann’s conjecture).
In this article a rather simple approach is presented, by means of which, is extended
from real to any complex number .
Theorem 10. (i) If , the Riemann’s zeta function can be expressed as
, (5-2)
or taking (1-5) into account,
. (5-3)
(ii) For real, (5-3) coincides with as originally defined in (5-1).
Proof: (i) As a starting point we note that as it appears in (5-1), with real , can be
obtained from in (4-1), if we set , i.e,
, real , (5-4)
and according to Theorem 7, can be expressed equivalently as,
real . (5-5)
However, by Theorem 8, the series in (5-5) converges for all , and thus provides
the extension of from real to any .
(ii) The proof follows easily, as an immediate consequence of (4-8).
The values of the extended Riemann’s zeta function , at negative integers, can now be
easily obtained, as shown in the following Theorem.
Theorem 11.(i) If and are the corresponding Bernoulli numbers, then
. (5-6)
(ii) In particular, , (5-7)
showing that the trivial zeros of , occur at .
Proof: (i) From (4-10) we have,
since , ([1]).
(ii) In particular , if ,
since for .
6. The Ser’s expression of the Riemann’s zeta function.
Helmut Hasse’s global expression for was published in 1930. Four years earlier, in 1926,
the French mathematician Joseph Ser, published the following (less known) expression for
, ([4], [14]),
(6-1)
Below we present a simple proof of Ser’s expression for .
Proof: From equation (5-2) we have,
(6-2)
Expanding the difference according to (3-6), and taking into account Theorem
12, (6-2) finally simplifies to
(6-3)
and making use of (1-5), equation (6-1) is readily obtained.
7. Appendix 1.
Theorem 12. If and , then . (7-1)
Proof: (i) We first consider the case where is a fixed real number.
If , the difference , for , and (6-1) is obviously
true. For all other values of , we know that and belong to , by Theorem 14, in
[10]. Let us consider the finite sum,
(7-2)
(by virtue of (3-6)), which eventually simplifies to,
, (7-3)
As , and since , , equation (6-3) implies,
, or
, from which
(ii) Let now , be a fixed complex number, with and .
From (3-12) , , and since as ,
and
, as proved in part (i), we get that .
(iii) We now consider the case , with and .
(a) For , we have from (3-19),
, for .
As ,
while , as proved in (i), so
(b) For , we have from (3-25),
, for .
As ,
while , as proved in part (i), and therefore
.
(c) The generalization is now obvious, and thus Theorem 12, is eventually proved for all
complex values of .
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