Abstract
Let \(\lambda _{f}(n),\sigma (n)\) and \(\varphi (n)\) be the nth Hecke eigenvalue of the normalized Hecke–Maass cusp form, the sum-of-divisors function and the Euler totient function, respectively. We investigate the asymptotic behaviour of the following summatory function:
as \(x\rightarrow \infty \), where \(1\leqslant j\leqslant 4\) is any fixed integer.
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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, for thoroughly reporting countless 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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This work was financially supported in part by the National Key Research and Development Program of China (Grant No. 2021YFA1000700) and Natural Science Basic Research Program of Shaanxi (Program Nos. 2023-JC-QN-0024, 2023-JC-YB-077).
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Hua, G. Mean value estimate of a hybrid arithmetic function attached to Fourier coefficients of Hecke–Maass forms. European Journal of Mathematics 9, 30 (2023). https://doi.org/10.1007/s40879-023-00631-2
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DOI: https://doi.org/10.1007/s40879-023-00631-2