Remark on the Hurwitz-Lerch zeta function
© Choi; licensee Springer. 2013
Received: 21 January 2013
Accepted: 7 March 2013
Published: 26 March 2013
Various generalizations of the Hurwitz-Lerch zeta function have been actively investigated by many authors. Very recently, Srivastava presented a systematic investigation of numerous interesting properties of some families of generating functions and their partial sums which are associated with various classes of the extended Hurwitz-Lerch zeta functions. In this paper, firstly, we show that by using the Poisson summation formula, the analytic continuation of the Lerch zeta function can be explained and the functional relation for the Lerch zeta function can be obtained in a very elementary way. Secondly, we present another functional relation for the Lerch zeta function and derive the well-known functional relation for the Hurwitz zeta function from our formula by following the lines of Apostol’s argument.
MSC: 11M99, 33B15, 42A24, 11M35, 11M36, 11M41, 42A16.
Keywordsgamma function Riemann zeta function generalized (or Hurwitz) zeta function Poisson summation formula Lerch zeta function Hurwitz-Lerch zeta function
1 Introduction and preliminaries
It is noted that, for convenience, in (1.2) is denoted simply by throughout this paper.
which, by means of (1.3), is a meromorphic function with only a simple pole at . For nonintegral x, becomes an entire function s.
and using Cauchy’s theorem in connection with the contour integral (1.4).
by making a basic use of the transformation theory of theta-functions. Apostol  noted that his proof is of particular interest because the usual approach (Riemann’s second method ) does not lead to the functional equation (1.6) and carried his method through to obtain (1.6) with further properties of , having no analogue in the case of .
which, upon setting , yields the Riemann functional equation (1.7).
Mordell  see also  proved the functional relation for in (1.10) and that can be continued meromorphically to the whole s plane except for a simple pole at , in a very elementary way, by using Poisson’s summation formula. Apostol  showed that (1.10) could be derived from (1.6) by giving an elaborate argument.
Here we aim mainly at, first, showing that in (1.2) becomes an entire function of s for nonintegral and the functional relation for can be obtained by using Poisson’s summation formula in a very elementary way; and, secondly, deriving the relations (1.11) and (1.12) and showing how the functional relation (1.10) for the Hurwitz zeta function can be obtained from (1.12) by just following Apostol’s arguments  and , respectively.
2 An analytic continuation of
If x is an integer, then in (1.2) reduces to the Hurwitz zeta function in (1.5). Since for all , throughout this argument, it is supposed that is defined at , (x being fixed) and .
where the prime ′ denotes that is omitted from the summation. The summation on the left refers to the integral values of n given by ; but, when either α or β is an integer, the corresponding term is halved. On the right, the summation means . It is supposed that
and are continuous in , the obvious one-sided continuity only being required at or ;
and are such that the integrals and converge, and is an integral of .
If either α or β is infinite, say , further condition is required that and as .
It is also observed that the integral in (2.5) now converges , and so is analytic for . Continuing in this way, it is found that can be continued to an entire function of s.
It is noted that the last integral in (2.8) converges for , and the second summation in (2.6) converges for . Therefore in (2.6) is analytic for .
The integral in (2.9) converges for . The expression of in (2.9), proved for , shows that is analytic for since then the general term is uniformly in s for s bounded and for every . Employing integration by parts repeatedly, we observe that can be continued analytically to the whole s plane.
which, upon replacing s by , yields (1.6).
3 Proof of (1.11) and (1.12)
we are led to the relation (1.8). Instead, replacing s by , x by −a, a by x in (3.7) as they were in [, p.163], we obtain the desired identity (1.11).
Adding (1.11) to (3.10), we are led to the desired relation (1.12).
4 (1.10) can be deduced from (1.12)
where the empty sum is understood to be nil.
Now, by repeating Apostol’s argument, it can be shown that vanishes identically for .
Dedicated to Professor Hari M Srivastava.
The author would like to express his deep gratitude to the referees who read through this paper in a detailed and critical manner to point out some misprints. This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0011005).
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