TY - JOUR

T1 - The Lerch zeta function I. Zeta integrals

AU - Lagarias, Jeffrey C.

AU - Winnie Li, Wen Ching

N1 - Funding Information:
The research of the first author was supported by NSF grant DMS-0500555 and DMS-0801029 and that of the second author by NSF grant DMS-0457574 and DMS-0801096.

PY - 2012/1

Y1 - 2012/1

N2 - This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions, and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a, c) with 0 < a; c < 1. It extends these functions to real a and c, and studies limiting cases of these functions where at least one of a and c take the values 0 or 1. A main feature is that as a function of three variables (s, a, c), in which a and c are real, the Lerch zeta function has discontinuities at integer values of a and c. For fixed s, the function ζ(s, a, c) is discontinuous on part of the boundary of the closed unit square in the (a, c)-variables, and the location and nature of these discontinuities depend on the real part R(s) of s. Analysis of this behavior is used to determine membership of these functions in L p([0, 1] 2; da dc) for 1 ≤ p < ∞, as a function of R(s). The paper also defines generalized Lerch zeta functions associated to the oscillator representation, and gives analogous four-term functional equations for them.

AB - This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions, and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a, c) with 0 < a; c < 1. It extends these functions to real a and c, and studies limiting cases of these functions where at least one of a and c take the values 0 or 1. A main feature is that as a function of three variables (s, a, c), in which a and c are real, the Lerch zeta function has discontinuities at integer values of a and c. For fixed s, the function ζ(s, a, c) is discontinuous on part of the boundary of the closed unit square in the (a, c)-variables, and the location and nature of these discontinuities depend on the real part R(s) of s. Analysis of this behavior is used to determine membership of these functions in L p([0, 1] 2; da dc) for 1 ≤ p < ∞, as a function of R(s). The paper also defines generalized Lerch zeta functions associated to the oscillator representation, and gives analogous four-term functional equations for them.

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U2 - 10.1515/FORM.2011.047

DO - 10.1515/FORM.2011.047

M3 - Article

AN - SCOPUS:84858673541

SN - 0933-7741

VL - 24

SP - 1

EP - 48

JO - Forum Mathematicum

JF - Forum Mathematicum

IS - 1

ER -