On conjectures of Sato–Tate and Bruinier–Kohnen

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Arias-de-Reyna S., İNAM İ., Wiese G.

Ramanujan Journal, vol.36, no.3, pp.455-481, 2015 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 36 Issue: 3
  • Publication Date: 2015
  • Doi Number: 10.1007/s11139-013-9547-2
  • Journal Name: Ramanujan Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.455-481
  • Keywords: Density of sets of primes, Fourier coefficients of modular forms, Half-integral weight modular forms, Sato–Tate equidistribution, Shimura lift
  • Bilecik Şeyh Edebali University Affiliated: Yes

Abstract

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind–Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato–Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.