Abstract
This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the $(a, c)$-variables having the Lerch zeta function as an eigenfunction.
Highlights
The Lerch zeta function is defined by the series ζ (s, a, c) ∞ n=0e2π ina (n + c)s (1)which absolutely converges for complex variables (s) > 1, (c) > 0 and (a) ≥ 0
(1) We introduce a set of auxiliary operators acting on twisted-periodic function spaces, a linear partial differential operator DL, a unitary operator R and the family of twovariable Hecke operators then have m (Tm), viewed as acting on the unit square through the use of twisted-periodic function spaces
This paper studied two-variable Hecke operators Tm given by (6) on spaces of functions of two real variables
Summary
Which absolutely converges for complex variables (s) > 1, (c) > 0 and (a) ≥ 0. The question this paper considers is that of obtaining an extension of the operators above inside suitable function spaces, in which the Lerch zeta function will be a simultaneous eigenfunction for all s ∈ C. It suffices to study these functions inside the unit square, since twisted-periodicity uniquely extends the function to R × R, providing a means of defining the two-variable Hecke operator action on R × R using only function values defined inside the unit square Another important ingredient of our extension is an operator encoding the functional equations. 6, we show that for each s ∈ C there is a twodimensional vector space Es of simultaneous eigenfunctions, the Lerch eigenspace, satisfying suitable integrability side conditions This is a generalization of Milnor’s converse result characterizing the Hurwitz zeta function and Kubert functions. The operator DL has features suggested for a putative “Hilbert– Polya” operator
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