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On the Distribution of Hecke Eigenvalues over Piatetski-Shapiro Prime Twins

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Abstract

Let λf (n) be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and ℙc(x): = {p ≤ x ∣ [pc]prime}, c ∈ ℝ+. In this paper, we show that for all 0 < c < 1 the mean value of λf(n) in ℙc(x) is ≪x log−Ax assuming the Riemann Hypothesis. Unconditionally, in the sense of Lebesgue measure, it holds for almost all c ∈ (ε, 1 − ε).

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Acknowledgements

The authors would like to express their gratitude to the referee for his or her careful reading and valuable suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, P. R. China

    De Yu Zhang

  2. Department of Mathematics, China University of Mining and Technology, Beijing, 100083, P. R. China

    Wen Guang Zhai

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  1. De Yu Zhang
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  2. Wen Guang Zhai
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Correspondence to De Yu Zhang.

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Supported by National Natural Science Foundation of China (Grant Nos. 11771256, 11971476)

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Zhang, D.Y., Zhai, W.G. On the Distribution of Hecke Eigenvalues over Piatetski-Shapiro Prime Twins. Acta. Math. Sin.-English Ser. 37, 1453–1464 (2021). https://doi.org/10.1007/s10114-021-0174-3

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  • DOI: https://doi.org/10.1007/s10114-021-0174-3

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