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

Abstract

We prove that more than nine percent of the central values L.1=2; p / are non-zero, where p Á 1 .mod 8/ ranges over primes and 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; 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.

Nonvanishing of hyperelliptic zeta functions over finite fields

“…This result improves on [3, Corollary 2.6] of Andrade and Keating and on [2, Corollary 2.8] of Andrade, Bae, and Jung, which give a proportion on order (log X) −2 , and goes beyond the methods of [1] by Andrade and Baluyot. For the analogous problem over Z, we refer to the recent work [4] of Baluyot and Pratt.…”

Lower bounds for negative moments of quadratic Dirichlet $L$-functions

“…• is the Kronecker symbol. In [2], S. Baluyot and K. Pratt proved that more than 9% of the members of the quadratic family of Dirichlet L-functions with prime moduli do not vanish at the central point.…”

“…Subsequent improvements on the error terms concerning these moments can be found in [14,32,33,[35][36][37]. An asymptotic formula for the second moment with prime moduli was obtained by S. Baluyot and K. Pratt [2] under GRH.…”

“…0 ≤ k ≤ 2) unconditionally, while sharp upper bounds are obtained for all real k > 2 under GRH. For the family with prime moduli or prime-related moduli, sharp upper and lower bounds were obtained by S. Baluyot and K. Pratt [2] for the second unconditionally and third moment under certain assumptions, by P. Gao and L. Zhao [13] for all k-th moment with k ≥ 0 under GRH.…”

Vanishing of Dirichlet L-functions at the central point over function fields

“…The behavior of these L-functions in the critical strip {s ∈ C|0 < ℜ(s) < 1}, specifically the locations of the zeroes of L(s, χ), has been a subject of intense study in number theory. In particular, under the Generalized Riemann Hypothesis (GRH), it is expected that the only points in the critical strip where L(s, χ) vanishes are on the vertical line ℜ(s) = 1 2 . In a different direction, a conjecture of Chowla [3] predicts that the L function L(s, χ) associated to a quadratic character χ does not vanish for any real s ∈ (0, 1), and in particular L( 1 2 , χ) = 0 for any χ.…”