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

TL;DR

This paper proves that over 9% of the central values of certain quadratic Dirichlet L-functions are non-zero for primes congruent to 1 mod 8, establishing a positive proportion and analyzing moments of these values.

Contribution

It demonstrates a positive lower bound on the proportion of non-zero central L-values for quadratic characters of prime conductor, a previously unknown result.

Findings

More than 9% of these L-values are non-zero.

Established the order of magnitude of the second moment.

Conditional results on the third moment and asymptotic sharpness under GRH.

Abstract

We prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $\chi_p$ 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 $L(\frac{1}{2},\chi_p)$, 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.