Counting zeros of Dirichlet L-functions

2020/05/06 by Michael A. Bennett, Bennett, Michael A., Greg Martin +5 · 5 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Algebraic Geometry and Number Theory

paper · doi:10.1090/mcom/3599

Abstract

We give explicit upper and lower bounds for N(T,χ ), the number of zeros of a Dirichlet L-function with character χ and height at most T. Suppose that χ has conductor q>1, and that T≥ 5/7. If ℓ =log \frac q(T+2)2π > 1.567, then | N(T,χ ) - ( \frac Tπ log \frac qT2π e -\frac χ (-1)4 ) | ≤ 0.22737 ℓ + 2 log (1+ℓ ) - 0.5. We give slightly stronger results for small q and T. Along the way, we prove a new bound on |L(s,χ )| for σ <-1/2.

Citations

• An improved upper bound for the argument of the Riemann zeta-function on the critical line II

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• A central limit theorem for partitions involving generalised divisor functions
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