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 $L(\frac{1}{2},{\chi }_p)$ 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 $L(\frac{1}{2},{\chi }_p)$, 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.

77 pages