Zeros near s=1 and the constant term of L'/L for L-functions in the Selberg class

2020/01/08 by Christian Táfula, Táfula, Christian
Mathematics · #Analytic Number Theory Research #Mathematical Approximation and Integration #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2001.02405

Abstract

Let L(s) = ∑_n=1^∞ a_n n^-s be an L-function in the Selberg class, and q_L its conductor. Let ℓ₀(L) be the constant term of the Laurent expansion of L'/L at s=1. We show that for certain families F of L-functions in the Selberg class with polynomial Euler product: \bullet If L\inF has no zeros β+ iγ with β> 1 - δ(log q_L)^-1, |γ| < (log q_L)^-1/2 for some absolute δ>0, then \Re(ℓ₀(L)) ≪_F log q_L; \bullet If \Re(ℓ₀(L)) ≪ log q_L for all L∈ F, then there is some absolute δ> 0 such that L has no zeros β+ iγ with β> 1 - δ(log q_L)^-1, |γ| < (1-β)^1/2(log q_L)^-1/2. This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.

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