On conjectures of Sato–Tate and Bruinier–Kohnen
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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.
KeywordsHalf-integral weight modular forms Shimura lift Sato–Tate equidistribution Fourier coefficients of modular forms Density of sets of primes
Mathematics Subject Classification11F37 11F30 11F80 11F11
The authors would like to thank Juan Arias de Reyna for his remarks. They also thank Jeremy Rouse for explanations concerning . I.I. and G.W. are grateful to Winfried Kohnen for interesting discussions. Thanks are also due to the anonymous referee for helpful suggestions concerning the presentation of the paper.
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