Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues (Q6562933)

Let \(\{\mathcal{A}_{F}(m,1,\dots,1)\}\) be the normalized Fourier coefficients of Hecke-Maass forms for \(\mathrm{SL}(n,\mathbb{Z})\). In the paper under review, the author proves conditionally, under assuming the generalized Ramanujan-Petersson conjecture and the \(\mathrm{SL}(n,\mathbb{Z})\) Vinogradov-Korobov zero-free regions, that if \(X\) is sufficiently large and \(e^{ (\log X)^{1-\varepsilon}} \ll_{\varepsilon} h_{1} \leq h_{2} \ll_{\varepsilon} X^{1-\varepsilon}\) where \(\varepsilon>0\) is fixed small, then\N\[\N\frac{1}{X} \int_{X}^{2X} \left|\frac{1}{h_{1}}\sum_{m=x}^{x+h_{1}} \mathcal{A}_{F}(m,1,\dots,1)- \frac{1}{h_{2}}\sum_{m=x}^{x+h_{2}} \mathcal{A}_{F}(m,1,\dots,1) \right|^{2}dx \ll_{F,\varepsilon} (\log X)^{-2/3+\varepsilon}.\N\]\NAs an application he shows that for \(f\) a Hecke cusp form and its eigenvalues \(\lambda_{f}(n)\) of the \(n\)th Hecke operator, if \(h,X\) are integers satisfying \(1 \leq h \ll X^{2/3}\), then the following approximation holds for almost all \(x \in [X,2X]\),\N\[\N\frac{1}{h} \left|\sum_{n=x}^{x+h} \lambda_{f}(n)\right| \ll_{f,\varepsilon} \max\Big(h^{-\frac{1}{2}}, X^{-\frac{1}{6}+\varepsilon} \Big).\N\]\NThe author also obtains non-trivial upper bounds for shifted convolution sums involving \(\{\mathcal{A}_{F}(m,1,\dots,1)\}\).