Abstract
Let \(\lambda _\pi (1,\ldots ,1,n)\) be the normalized Fourier coefficients of an even Hecke–Maass form \(\pi \) for \(SL(m, {\mathbb {Z}})\) with \(m\ge 3\), and \(r_{3}(n)=\#\{(n_1,n_2,n_3)\in {\mathbb {Z}}^3:n=n_1^2+n_2^2+n_3^2\}\). In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients \(\lambda _\pi (1,\ldots ,1 ,n)\) and \(r_{3}(n)\), which improves previous results (even under the generalized Ramanujan conjecture).
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Acknowledgements
The authors are very grateful to the referee for some extremely helpful remarks. In particular, we are indebted to the referee for pointing out a gap in the proof of Lemma 4.6 in an earlier version of the paper. The first named author would like to thank Yujiao Jiang for his encouragement and advice.
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This work is supported in part by NSFC (Grant Nos. 11771252, 12031008), IRT16R43, and the Taishan Scholar Project.
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Hu, G., Lü, G. Shifted convolution sums of GL(m) cusp forms with \(\Theta \)-series. Ramanujan J 56, 555–584 (2021). https://doi.org/10.1007/s11139-021-00447-2
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DOI: https://doi.org/10.1007/s11139-021-00447-2