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

  title={Dirichlet \$L\$-functions of quadratic characters of prime conductor at the central point},
  author={Siegfred Alan C. Baluyot and Kyle Pratt},
  journal={Journal of the European Mathematical Society},
  • S. Baluyot, Kyle Pratt
  • Published 26 September 2018
  • Mathematics
  • Journal of the European Mathematical Society
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… Expand
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