JournalsjemsVol. 24, No. 2pp. 369–460

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

  • Siegfred Baluyot

    University of Illinois at Urbana-Champaign, USA
  • Kyle Pratt

    University of Illinois at Urbana-Champaign, USA
Dirichlet $L$-functions of quadratic characters of prime conductor at the central point cover
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Abstract

We prove that more than nine percent of the central values L(1/2,χp)L(1/2,\chi_p) are non-zero, where p1(mod8)p\equiv 1 \pmod{8} ranges over primes and χp\chi_p is the real primitive Dirichlet character of conductor pp. 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,χp)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.

Cite this article

Siegfred Baluyot, Kyle Pratt, Dirichlet LL-functions of quadratic characters of prime conductor at the central point. J. Eur. Math. Soc. 24 (2022), no. 2, pp. 369–460

DOI 10.4171/JEMS/1084