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