Asymptotics of a Gauss hypergeometric function related to moments of symmetric-square $L$-functions II

TL;DR

This paper derives an asymptotic formula for a specific Gauss hypergeometric function relevant to the first moment of symmetric-square L-functions of Maass forms, as parameters grow large and their ratio approaches zero.

Contribution

It provides the first asymptotic analysis of this hypergeometric function in the specified limit, connecting special function behavior to number theory.

Findings

Asymptotic formula for the hypergeometric function as r,t→∞ with r/t→0

Application to moments of Maass form symmetric-square L-functions

Enhanced understanding of special functions in analytic number theory

Abstract

We prove an asymptotic formula for $F\left(1/4-it+ir,1/4-it-ir,1/2;x \right)$ as $r, t\to\infty$ and $\alpha=r/t\to0.$ This special case of the Gauss hypergeometric function appears in the explicit formula for the first moment of Maass form symmetric-square $L$-functions.