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A short note on the Bruiner–Kohnen sign equidistribution conjecture and Halász’ theorem

Department of Mathematics, Bilecik Seyh Edebali University, Bilecik 11020, Turkey

E-mail Address: ilker.inam@bilecik.edu.tr

and

Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg

E-mail Address: gabor.wiese@uni.lu

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https://doi.org/10.1142/S1793042116500214Cited by:5 (Source: Crossref)

Abstract

In this note, we improve earlier results towards the Bruinier–Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halász’ theorem. Moreover, by applying a result of Serre, we remove all unproved assumptions.

Keywords:

AMSC: 11F37, 11F30

References

1. S. Arias-de-Reyna, I. Inam and G. Wiese, On the conjectures of Sato–Tate and Bruinier–Kohnen, Ramanujan J. 36, (2015) 455–481. Crossref, Web of Science, Google Scholar
2. T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, A family of Calabi–Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011) 29–98. Crossref, Web of Science, Google Scholar
3. G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hung. 19 (1968) 365–403. Crossref, Web of Science, Google Scholar
4. I. Inam and G. Wiese, Equidistribution of signs for modular eigenforms of half integral weight, Arch. Math. 101 (2013) 331–339. Crossref, Web of Science, Google Scholar
5. K. Matomäki and M. Radziwill, Sign changes of Hecke eigenvalues, to appear in Geom. Funct. Anal.; arXiv:1405.7671. Google Scholar
6. J. Rouse and J. Thorner, The explicit Sato–Tate conjecture and densities pertaining to Lehmer type questions, to appear in Trans. Amer. Math. Soc.; arXiv:1305.5283. Google Scholar
7. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci. 54 (1981) 123–201. Crossref, Google Scholar
8. G. Shimura, On modular forms of half-integral weight, Ann. Math. 97 (1973) 440–481. Crossref, Web of Science, Google Scholar