Dirichlet L-functions of quadratic characters of prime conductor at the central point

2018/09/26 by Siegfred Baluyot, Kyle Pratt, Baluyot, Siegfred +1 · 2 citations
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1809.09992

77 pages

arxiv created 2018/09/26 · arxiv updated 2018/09/27

Abstract

We prove that more than nine percent of the central values L((1)/(2),χ_p) are non-zero, where p≡ 1 \pmod8 ranges over primes and χ_p is the real primitive Dirichlet character of conductor p. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of L((1)/(2),χ_p), and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.

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• Decorrelation of SO(5)× SO(2) Bessel periods for symmetric cubes of GL(2)
• Selberg orthogonality for half-integral weight modular forms

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• On the Moments of Certain Families of Dirichlet L-functions
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• Small zeros of Dirichlet L-functions of quadratic characters of prime modulus
• Explicit Upper Bounds for |L(1, χ)| when χ(3)=0