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

2015/10/31 by Dahl, Alexander
#11F68 (Secondary) #11M32 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1511.00071

Abstract

We study a double Dirichlet series of the form ∑_dL(s,χ_dχ)χ'(d)d^-w, where χ and χ' 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 ℂ². A convexity bound at the central point is established to be (MN)^3/8+ε and a subconvexity bound of (MN(M+N))^1/6+ε is proven. The developed theory is used to prove an upper bound for the smallest positive integer d such that L(1/2,χ_dN) does not vanish, and further applications of subconvexity bounds to this problem are presented.

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