On averages of Fourier coefficients of Maass cusp forms (Q1945793)

The author investigates sums over the Fourier coefficients of primitive Maass cusp forms. When \(\varphi\) is a primitive Maass cusp form and \(t_{\varphi}(n)\) its \(n\)th Fourier coefficient at the cusp infinity, then for any \(\varepsilon>0\) he achieves the following two results: \[ \sum_{n\leq x}t_{\varphi}(n)\ll_{\varphi,\varepsilon}x^{\frac{1027}{2827}+\varepsilon}\quad\text{and}\;\sum_{n\leq x}t_{\varphi}(n^{2})\ll_{\varphi,\varepsilon}x^{\frac{489}{861}+\varepsilon}, \] thus improving the fairly investigated earlier scientific records. Moreover, as a byproduct of concluding the latter result, he ends up with the following: when \(\text{sym}^{2}\varphi\) is the symmetric-square lift of a primitive Maass cusp form \(\varphi\) and \(t_{\varphi}(m,n)\) is its Fourier coefficient, then for any \(\varepsilon>0\) \[ \sum_{n\leq x}t_{\varphi}(1,n)\ll_{\varphi,\varepsilon}x^{\frac{489}{861}+\varepsilon}. \] The proofs are leaning on combining the results by \textit{H. H. Kim} and \textit{F. Shahidi} [Ann. Math. (2) 155, No. 3, 837--893 (2002; Zbl 1040.11036), Duke Math. J. 112, No. 1, 177--197 (2002; Zbl 1074.11027)] with the ones by \textit{K. Chandrasekharan} and \textit{R. Narasimhan} [Ann. Math. (2) 76, 93--136 (1962; Zbl 0211.37901)].