Subconvexity for a double Dirichlet series and non-vanishing of $L$-functions

TL;DR

This paper establishes subconvexity bounds for a double Dirichlet series, extending the understanding of L-functions and providing bounds on non-vanishing at the central point, with implications for number theory.

Contribution

It introduces new subconvexity bounds for a specific double Dirichlet series and applies these bounds to non-vanishing results of L-functions at the central point.

Findings

Proves a convexity bound of (MN)^{3/8+ε}.

Establishes a subconvexity bound of (MN(M+N))^{1/6+ε}.

Provides an upper bound for the smallest non-vanishing d.

Abstract

We study a double Dirichlet series of the form $\sum_{d}L(s,\chi_{d}\chi)\chi'(d)d^{-w}$, where $\chi$ and $\chi'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to $\mathbb{C}^{2}$. A convexity bound at the central point is established to be $(MN)^{3/8+\varepsilon}$ and a subconvexity bound of $(MN(M+N))^{1/6+\varepsilon}$ is proven. The developed theory is used to prove an upper bound for the smallest positive integer $d$ such that $L(1/2,\chi_{dN})$ does not vanish, and further applications of subconvexity bounds to this problem are presented.