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On conjectures of Sato–Tate and Bruinier–Kohnen

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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.

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Notes

  1. Added in proof: The notion of regular set of primes already appeared in [4].

  2. Added in proof: The assumption of the Riemann Hypothesis is not necessary, see [19], Proposition 1.5.

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Acknowledgements

The authors would like to thank Juan Arias de Reyna for his remarks. They also thank Jeremy Rouse for explanations concerning [18]. 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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Correspondence to Ilker Inam.

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I.I. is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) and Uludag University Research Project No: UAP(F) 2012/15. G.W. acknowledges partial support by the priority program 1489 of the Deutsche Forschungsgemeinschaft (DFG). S.A. is partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain. I.I. would like to thank the University of Luxembourg for its hospitality.

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Arias-de-Reyna, S., Inam, I. & Wiese, G. On conjectures of Sato–Tate and Bruinier–Kohnen. Ramanujan J 36, 455–481 (2015). https://doi.org/10.1007/s11139-013-9547-2

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