A Product Identity For Dirichlet Series Satisfying Hecke's Functional Equation

TL;DR

The paper develops a Wilton-type product identity for Dirichlet series that satisfy Hecke's functional equation, unifying a broad family of L-functions under a single analytic framework. The approach uses a Perron-type summation operator, Meijer $G$ representations, and contour-residue methods to express $\varphi(u)\psi(v)$ in terms of residues at $s=k$ and Bessel-integral corrections, with lemmas translating products into Dirichlet convolution sums. The results yield explicit, computable product identities for Hecke series, modular L-functions, Ramanujan's $L$-function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields, and Dirichlet $L$-functions, including a notable 4-term identity for $\zeta(s)$. This framework clarifies how automorphic $L$-functions interact under multiplicative convolution and provides practical representations for evaluating special values and mean-type quantities across a broad spectrum of zeta and $L$-functions.

Abstract

In this paper, we give an analogue of Wilton's product formula for Dirichlet series that satisfy Hecke's functional equation. We apply our results to obtain identities for Hecke series, L-functions associated to modular forms, Ramanujan's L-function, Epstein zeta functions, Dedekind zeta functions of imaginary quadratic fields and Dirichlet L-functions. A $4$-term product identity for Riemann zeta function is also given.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 4.1
• Proposition 4.2
• Proposition 4.3