Abstract
In this paper, we solve the Rankin–Selberg problem. That is, we break the well known Rankin–Selberg’s bound on the error term of the second moment of Fourier coefficients of a \({\text {GL}}(2)\) cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.
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Notes
There is a typo in [14, eq. (20.158)].
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Acknowledgements
The author wants to thank Prof. Jianya Liu and Zeév Rudnick for their help and encouragements. He would like to thank Yongxiao Lin and Qingfeng Sun for many discussions on [5, 21]. The author started to try seriously to solve the Rankin–Selberg problem when he was a postdoc at Tel Aviv University. He likes to thank Tel Aviv Unviersity for providing a nice work environment.
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Communicated by Kannan Soundararajan.
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This work was supported by the Young Taishan Scholars Program of Shandong Province (Grant No. tsqn201909046), Qilu Young Scholar Program of Shandong University, and NSFC (Nos. 12001314 and 12031008)
