Non-vanishing of certain integral representations

TL;DR

This paper demonstrates the existence of specific functions for which certain integral representations do not vanish across all complex parameters, providing insights into the poles of related L-functions.

Contribution

It establishes the non-vanishing of local Flicker and Bump-Friedberg integrals for all complex parameters, revealing new properties of these integral representations.

Findings

Existence of non-vanishing Whittaker and Schwartz functions for all complex s

Non-vanishing of local Bump-Friedberg integrals for all complex pairs (s1,s2)

Identification of potential pole locations for associated partial L-functions

Abstract

In this paper, we prove that there exist Whittaker and Schwartz functions such that the local Flicker integrals are non-vanishing for all complex values of $s$, and the local Bump-Friedberg integrals are non-vanishing for all complex pairs $(s_1,s_2)$. As a corollary, we determine the potential locations of poles for their corresponding partial $L$-functions.