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

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 χp(·) = (·|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 ofÖzlük and Snyder, we prove that GRH implies L( 1 2 , χp) = 0 for at least 75% of the primes.

Hilbert spaces and low-lying zeros of L-functions

“…There are many examples of families of L-functions in the literature which are amenable to our framework. A non-exhaustive list includes, for instance, the works [2,3,4,9,12,13,15,16,17,20,22,23,24,26,27,28,30,36,38,39,40,41,43,46]. In order to keep the exposition short, the reader is invited to consult the corresponding excerpts from these works for the precise definitions and technicalities, and we only briefly comment on a few of these examples below.…”

“…However, when ∆ " 2 we are able to show that a large proportion of this family cannot vanish in certain ranges near central point (the zeros are repelled from s " 1 2 ). For instance, Figure 4 shows that, for every t P r 3 10 , 4 10 s, at least 70% of the f P H ḱ pN q have L `1 2 `2πit log k 2 N , f ˘‰ 0 as N Ñ 8 over square-free integers. 3) holds for this family with G " Sp for ∆ " 1 and, in [20, Theorem 1.2], with an additional restricted average over D, they are able to prove that this larger family is symplectic with extended support ∆ " 4{3.…”

One level density of low-lying zeros of quadratic Hecke L-functions to prime moduli

“…We point out here that our formulation of the 1-level density is the more commonly used one in the literature, while in [AB20], the 1-level density is formed using a form factor, as initially used bÿ Ozlük and Snyder in [ÖS99]. Now we state our result on the one level density as follows.…”

Moments of Dirichlet L–functions with prime conductors over function fields

“…Using sieve methods, they also showed that more than 9% of L(1/2, χ p ) are non-zero. Under GRH, Andrade and Baluyot [AB18] computed the 1-level density in the family and obtained that more than 75% of the L-functions evaluated at the central point do not vanish. The corresponding problem of computing moments in the family of quadratic Dirichlet L-functions with prime conductor over function fields was considered by Andrade and Keating [AK13].…”