The second and fourth moments of theta functions at their central point

Stéphane R. Louboutin, Marc Munsch

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Let χ range over the (p - 1)/2 even Dirichlet characters mod p ≥ 3, a prime. Let θ(x, χ) be the associated theta series. It is known that the square mean value of θ(1, χ) is asymptotic to p3/2/42 as p goes to infinity. We prove that the fourth mean value of θ(1, χ) is asymptotic to 3/16πp2logp as p goes to infinity. We give similar results for mean values of odd Dirichlet characters mod p.

Original languageEnglish
Pages (from-to)1186-1193
Number of pages8
JournalJournal of Number Theory
Issue number4
Publication statusPublished - 1 Apr 2013
Externally publishedYes



  • Dirichlet characters
  • Mean values
  • Theta functions

ASJC Scopus subject areas

  • Algebra and Number Theory

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