Small zeros of Dirichlet L-functions of quadratic characters of prime modulus

2018/02/09 by Andrade, Julio, Baluyot, Siegfred
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1802.03413

Abstract

In this paper, we investigate the distribution of the imaginary parts of zeros near the real axis of Dirichlet L-functions associated to the quadratic characters χ_p(⋅)=(⋅ |p) with p a prime number. Assuming the Generalized Riemann Hypothesis (GRH), we compute the one-level density for the zeros of this family of L-functions under the condition that the Fourier transform of the test function is supported on a closed subinterval of (-1,1). We also write down the ratios conjecture for this family of L-functions a la Conrey, Farmer and Zirnbauer and derive a conjecture for the one-level density which is consistent with the main theorem of this paper and with the Katz-Sarnak prediction and includes lower order terms. Following the methods of Özlük and Snyder, we prove that GRH implies L((1)/(2),χ_p)≠ 0 for at least 75% of the primes.

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