Subconvexity for a double Dirichlet series

Abstract

For Dirichlet series roughly of the type $Z(s,w)=su{m}_{d}L(s,ch{i}_d)d^{-w}$ the subconvexity bound $Z(s,w)\ll (sw(s+w){)}^{1/6+\epsilon }$ 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 Lindelöf 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.

17 pages

References in corpus (1)

• The subconvexity problem for $\GL_{2}$

Cited by in corpus (5)

• Subconvexity for Half Integral Weight $L$-functions
• Double Dirichlet series and quantum unique ergodicity of weight 1/2 Eisenstein series
• First moment of central values of quadratic Hecke $L$-functions in the Gaussian field
• First Moment of Quadratic Hecke $L$-Functions with Lower Order Term
• Subconvexity of a double Dirichlet series over the Gaussian field