On mean values for the exponential sum of divisor functions (Q6594435)

Let \(k\ge 2\) be an integer. Let \(\tau_k(\cdot)\) be the \(k\)-fold divisor function. So, for example, \(\tau_2(n)\) is the number of positive divisors of \(n\). In general, \(\tau_k(n)\) is the number of ways of writing \(n\) as the product of \(k\) positive integers. The author of the paper under review refines a previous result by \textit{M. Pandey} [C. R., Math., Acad. Sci. Paris 360, 419--424 (2022; Zbl 1505.11106)] on the mean values of certain exponential sums involving \(\tau_k\). The author's main result is Theorem 1.1 which states: For fixed \(s\) and \(k\), we have \N\[\N\int_0^1 \left|\sum_{1\le n\le x} \tau_k(n)e(\alpha n)\right|^2\, d\alpha = x^{s-1} (\log x)^{s(k-1)}\sum_{\ell \ge 0} \frac{\gamma_{\ell,s,k}}{(\log x)^{\ell}} + O(x^{s-1-\delta_{s,k}+\varepsilon}),\N\]\Nwhere \N\[\N\delta_{s,k} = \frac{2(s-2)}{(s+5/2)(k+1)+2},\N\]\Nthe coefficients \(\gamma_{\ell,s,k}\) satisfying the bound \(|\gamma_{\ell,s,k}|\le \exp\left(O_{s,k}(\ell)\right)\) for all \(\ell \ge 1\), \(\gamma_{0,s,k}>0\), and the implied constant may depend on \(s\), \(k\) and \(\varepsilon\).\N\NThe author therefore improves on the result by Pandey, published in 2022, who obtained a larger constant of \(7\) (rather than \(5/2\)) in the denominator of the power-saving term \(\delta_{s,k}\). The author's proof of this extra power-saving is short and is presented somewhat tersely, relying heavily on previous literature for various details. One will need to consult \textit{M. Pandey} [C. R., Math., Acad. Sci. Paris 360, 419--424 (2022; Zbl 1505.11106)] to complete the details in the proof of Proposition 2.1 by the author. As a side remark, Theorem 1.1 does not seem to state a lower bound on \(s\) but presumably such a lower bound is needed.