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- Publisher
- de Gruyter
- Copyright
- © 2012 by Walter de Gruyter Berlin Boston
- ISSN
- 0933-7741
- eISSN
- 1435-5337
- DOI
- 10.1515/form.2011.047
- Publisher site
- See Article on Publisher Site
Abstract
Abstract.This is the first of four papers that study algebraic andanalytic structures associated to the Lerch zeta function.This paper studies “zeta integrals” associated to the Lerch zeta functionusing test functions, and obtains functional equations for them. Special cases includea pair of symmetrized four-term functional equationsfor combinations of Lerch zeta functions,found by A. Weil, for real parameters (a,c)$(a,c)$ with 0<a,c<1$0< a, c< 1$.It extends these functions to real a$a$ and c$c$,and studies limiting cases of these functions where at least one of a$a$ and c$c$take the values 0 or 1. A main feature is thatas a function of three variables (s,a,c)$(s, a, c)$, in which a$a$ and c$c$ are real,the Lerch zeta function has discontinuities at integer values ofa$a$ and c$c$. For fixed s$s$,the function (s,a,c)$\zeta (s,a,c)$ is discontinuous on part of the boundary of theclosed unit square in the (a,c)$(a,c)$-variables, and thelocation and nature of these discontinuities depend onthe real part (s)$\Re (s)$ of s$s$. Analysis of this behavior is used to determinemembership of these functions in Lp([0,1]2,dadc)$L^p([0,1]^2, da\,dc)$for 1p<$1 \le p < \infty $, as a function of (s)$\Re (s)$. The paper also definesgeneralized Lerch zeta functions associated to the oscillator representation,and gives analogous four-term functional equations for them.
Journal
Forum Mathematicum – de Gruyter
Published: Jan 19, 2012
Keywords: Functional equation; Hurwitz zeta function; Lerch zeta function; periodic zeta function