Shifted convolution sums related to Hecke–Maass forms

Abstract

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:

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