Reciprocity for GL(2) L-functions twisted by Dirichlet characters

TL;DR

The paper proves an exact reciprocity relation for a first moment of GL(2) L-functions twisted by Dirichlet characters, effectively swapping key parameters between two twisted moments. It introduces a new, streamlined method based on additive reciprocity and functional equations to derive a finite-sum identity involving L-values at shifted half-integer arguments, applicable to holomorphic cusp forms and extendable to Maass forms with arbitrarily sharp error control. The results generalize prior work of Bettin, Drappeau, and Nordentoft to general twists (not just simple twists) and offer an exact formula in the holomorphic case along with a precise asymptotic expansion in the Maass case, improving the understanding of how moments of L-functions relate under parameter reciprocity. The approach relies on a careful combination of additive and multiplicative twists, Mellin-type transforms, and contour integration, avoiding more geometric or continued-fraction techniques and suggesting potential extensions to other GL(2) settings and higher levels.

Abstract

A formula connecting a moment of L-functions and a dual moment in a way that interchanges the roles of certain key parameters on both sides is known as a reciprocity relation. We establish a reciprocity relation for a first moment of GL(2) L-functions twisted by Dirichlet characters. This extends, via a new and simple argument, some results of Bettin, Drappeau, and Nordentoft.

Theorems & Definitions (23)

• Proposition 1
• Proposition 2
• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Lemma 2.1: Relating additive and multiplicative twists
• proof
• Lemma 2.2: Bounds on $L$-functions in the $t$-aspect