Nonvanishing for cubic L-functions

Abstract

We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$ . This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

Moments of quadratic Dirichlet $L$-functions over Function Fields

“…In applying the upper bounds method of Soundararajan-Harper, we shall also make crucial use of an upper bound on log |L(1/2, χ D )| for D ∈ H 2g+1,q in terms of a short Dirichlet polynomial over the primes. We now derive such a bound basing on the devices developed in the proofs of [1, Theorem 3.3], [13,Proposition 4.3] and [15,Lemma 3.1]. We recall from the above that L(s, χ D ) is a polynomial of degree at most 2g for D ∈ H 2g+1,q .…”

On the splitting conjecture in the hybrid model for the Riemann zeta function

“…Given a general Dirichlet polynomial of the form D(s) = p Y a(p)p −s , suppose t ∈ [T, 2T ] is such that |kD(s)| Z. Then by (17) we have (37) e kD(s) −…”

“…This circle of ideas has been used in a wide variety of different contexts recently. These include; short interval maxima of the Riemann zeta function [2,3,4], unconditional bounds for the moments of zeta and L-functions [21,26,27], value distribution of L-functions [16,30,39], sign changes in Fourier coefficients of modular forms [35], non-vanishing of central values of L-functions [17] and equidistribution of lattice points on the sphere [32]. In our case, we use these ideas to prove the following.…”

One-level density of the family of twists of an elliptic curve over function fields

“…For cubic Dirichlet L-functions over F q (t), David, Florea, and Lalín [DFL19] computed the mean values of the L-functions. They also proved that a positive proportion of these L-functions do not vanish at the central point [DFL20]. For twists of order r such that q ≡ 1 mod r, Meisner determined the number of F q -points of the associated curves [Mei15].…”