Shifted convolution sums related to Hecke–Maass forms


Let \(\phi (z)\) be a primitive Hecke–Maass cusp forms with Laplace eigenvalue \(\tfrac{1}{4}+t^2\). Denote by \(L(s, \mathrm{sym}^m\phi )\) the m-th symmetric power L-function associated to \(\phi \) and by \(\lambda _{\mathrm{sym}^m\phi }(n)\) the n-th coefficient of the Dirichlet expansion of \(L(s, \mathrm{sym}^m\phi )\). For any nonzero integer \(\ell \) we prove

$$\begin{aligned} \sum _{n\leqslant x} \left| \lambda _{\phi }(n)\lambda _{\phi }(n+\ell )\right| \ll _{\phi , \ell } \frac{x}{(\log x)^{0.187}} \qquad (x\geqslant 3). \end{aligned}$$

This improves Holowinsky’s corresponding result, which requires \(\tfrac{1}{6}\) in place of 0.187. for all \(x\geqslant 3\). Further assuming that \(L(s, \mathrm{sym}^{10}\phi )\) and \(L(s, \mathrm{sym}^{12}\phi )\) are automorphic cuspidal, we obtain a conditional generalization to the symmetric square case:

$$\begin{aligned} \sum _{n\leqslant x} \left| \lambda _{\mathrm{sym}^2\phi }(n)\lambda _{\mathrm{sym}^2\phi }(n+\ell )\right| \ll _{\phi , \ell } \frac{x}{(\log x)^{0.196}} \qquad (x\geqslant 3). \end{aligned}$$

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    In Sect. 3.5 below, we shall explain the reason behind these choices.


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This work was done during the visit of the first author to Institut Élie Cartan de l’Université de Lorraine. The hospitality and nice working conditions of IECL were gratefully acknowledged.

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Correspondence to Hengcai Tang.

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Hengcai Tang is supported by the National Natural Science Foundation of China (11871193) and Program for Young Scholar of Henan Province (2019GGJS026).

Jie Wu is supported by the National Natural Science Foundation of China (11771121, 11971370).

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Tang, H., Wu, J. Shifted convolution sums related to Hecke–Maass forms. Ramanujan J (2020).

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  • Fourier coefficients
  • Rankin–Selberg L-function
  • Sieve method

Mathematics Subject Classification

  • 11F30
  • 11F11
  • 11F66