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 $L(1/2,{\chi }_p)$ are non-zero, where $p\equiv 1(\mathrm{mod}8)$ 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 $L(1/2,{\chi }_p)$, 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.