Subconvexity for a double Dirichlet series

TL;DR

This paper establishes a subconvexity bound for a class of double Dirichlet series on the critical lines, improving upon the convexity bound, and also proves a mean square bound aligned with the Lindelöf hypothesis.

Contribution

It introduces a novel subconvexity bound for double Dirichlet series and provides a mean square bound consistent with the Lindelöf hypothesis, including a special case without an Euler product.

Findings

Proved subconvexity bound $Z(s,w) \\ll (sw(s+w))^{1/6+\\varepsilon}$ on critical lines.

Established a mean square bound compatible with the Lindelöf hypothesis.

Derived a subconvex bound for a Dirichlet series without an Euler product when $s=1/2$.

Abstract

For Dirichlet series roughly of the type $Z(s, w) = sum_d L(s, chi_d) d^{-w}$ the subconvexity bound $Z(s, w) \ll (sw(s+w))^{1/6+\varepsilon}$ is proved on the critical lines $\Re s = \Re w = 1/2$. The convexity bound would replace 1/6 with 1/4. In addition, a mean square bound is proved that is consistent with the Lindel\"of hypothesis. An interesting specialization is $s=1/2$ in which case the above result give a subconvex bound for a Dirichlet series without an Euler product.