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On the sign changes and non-vanishing of Hecke eigenvalues associated to symmetric power L-Functions

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Abstract

Let f be a Hecke cusp form of even integral weight k for the full modular group \(SL(2,{\mathbb {Z}})\). Denote by \(\lambda _{\text {sym}^{j}f}(n)\) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function associated with f. In this paper, we derive a upper bound for the first negative coefficient for sequence \(\{\lambda _{\text {sym}^{j}f}(n)\}_{n\ge 1}\) in terms of the weight of f. Furthermore, we also investigate the length of the maximal string of terms with the same sign in an interval [1, x].

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Acknowledgements

The author would like to express his gratitude to Professor Guangshi Lü and Professor Bin Chen for their constant encouragement, and valuable suggestions. The author is extremely grateful to the anonymous referees for their meticulous checking and thoroughly reporting many typos and inaccuracies as well as for their valuable comments. These corrections and additions have made the manuscript clearer and more readable.

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  1. School of Mathematics and Statistics, Weinan Normal University, Shaanxi, Weinan, 714099, China

    Guodong Hua

  2. School of Mathematics, Shandong University, Shandong, 250100, China

    Guodong Hua

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  1. Guodong Hua
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Correspondence to Guodong Hua.

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This work was supported in part by the National Key Research and Development Program of China (Grant No. 2021YFA 10000700)

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Hua, G. On the sign changes and non-vanishing of Hecke eigenvalues associated to symmetric power L-Functions. Ramanujan J 59, 775–789 (2022). https://doi.org/10.1007/s11139-022-00596-y

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