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Modular forms and an explicit Chebotarev variant of the Brun–Titchmarsh theorem

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Abstract

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that

$$\begin{aligned} \lim _{x \rightarrow \infty } \frac{\#\{1 \le n \le x \mid \tau (n) \ne 0\}}{x} > 1-1.15 \times 10^{-12}, \end{aligned}$$

where \(\tau (n)\) is Ramanujan’s tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that \(\tau (n) \ne 0\).

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Acknowledgements

The authors would like to thank Jesse Thorner for supervising this project and Ken Ono for his valuable suggestions. They are grateful for the support of grants from the National Science Foundation (DMS-2002265, DMS-2055118, DMS-2147273), the National Security Agency (H98230-22-1-0020), and the Templeton World Charity Foundation. This research was conducted as part of the 2022 Research Experiences for Undergraduates at the University of Virginia.

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  1. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Daniel Hu

  2. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Hari R. Iyer

  3. Department of Mathematics, Williams College, Williamstown, MA, USA

    Alexander Shashkov

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  1. Daniel Hu
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Correspondence to Hari R. Iyer.

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Hu, D., Iyer, H.R. & Shashkov, A. Modular forms and an explicit Chebotarev variant of the Brun–Titchmarsh theorem. Res. number theory 9, 46 (2023). https://doi.org/10.1007/s40993-023-00451-z

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