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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Akiyama, S., Tanigawa, Y.: Calculation of values of L-functions associated to elliptic curves. Math. Comput. 68(227), 1201–1231 (1999)
Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R.: A family of Calabi–Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011)
Bruinier, J.H., Kohnen, W.: Sign changes of coefficients of half integral weight modular forms. In: Edixhoven, B., van der Geer, G., Moonen, B. (eds.) Modular Forms on Schiermonnikoog, pp. 57–66. Cambridge University Press, Cambridge (2008)
Delange, H.: Sur la distribution des entiers ayant certaines propriétés. Ann. Sci. Éc. Norm. Super. 73(3), 15–74 (1956)
Delange, H.: Un théorème sur les fonctions arithmétiques multiplicatives et ses applications. Ann. Sci. Éc. Norm. Super. 78(3), 1–29 (1961)
Hecke, E.: Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Math. Z. 6(1–2), 11–51 (1920)
Inam, I., Wiese, G.: Equidistribution of signs for modular eigenforms of half integral weight. Arch. Math. 101(4), 331–339 (2013)
Korevaar, J.: The Wiener–Ikehara theorem by complex analysis. Proc. Am. Math. Soc. 134(4), 1107–1116 (2006)
Kohnen, W., Lau, Y.-K., Wu, J.: Fourier coefficients of cusp forms of half-integral weight. Math. Z. 273, 29–41 (2013)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)
Miyake, T.: Modular Forms. Springer, Berlin (2006). x+335 pp.
Murty, V.K.: Explicit formulae and the Lang–Trotter conjecture. Rocky Mt. J. Math. 15(2), 535–551 (1985)
Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, 3rd edn. Springer, Berlin (1990). 708 pp.
Niwa, S.: Modular forms of half integral weight and the integral of certain theta-functions. Nagoya Math. J. 56, 147–161 (1974)
Rajan, C.S.: Distribution of values of Hecke characters of infinite order. Acta Arith. 85, 279–291 (1998)
Ribet, K.A.: Galois Representations Attached to Eigenforms with Nebentypus, Modular Functions of One Variable, vol. V, pp. 17–51. Springer, Berlin (1977)
Rosser, B.: Explicit bounds for some functions of prime numbers. Am. J. Math. 63(1), 211–232 (1941)
Rouse, J., Thorner, J.: The explicit Sato–Tate conjecture and densities pertaining to Lehmer type questions. Preprint. arXiv:1305.5283
Serre, J.-P.: Divisibilité de certaines fonctions arithmétiques. Enseign. Math. (2) 22(3-4), 227–260 (1976)
Serre, J.P.: Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math. 54, 323–401 (1981)
Shimura, G.: On modular forms of half-integral weight. Ann. Math. 97, 440–481 (1973)
Tunnell, J.B.: A classical Diophantine problem and modular forms of weight 3/2. Invent. Math. 72, 323–334 (1983)
Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. 60, 375–484 (1981)
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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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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DOI: https://doi.org/10.1007/s11139-013-9547-2
Keywords
- Half-integral weight modular forms
- Shimura lift
- Sato–Tate equidistribution
- Fourier coefficients of modular forms
- Density of sets of primes