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arXiv · 1809.09992

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

Siegfred Baluyot·Kyle Pratt
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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.

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Siegfred Baluyot, Kyle Pratt. 2018-09-26. Dirichlet $L$-functions of quadratic characters of prime conductor at the central point. https://arxiv.org/abs/1809.09992

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