An asymptotic formula related to the sums of divisors (Q2833595)

Let \(d(n)\) denote the number of positive divisors of \(n\) and \(k\) be a positive integer. For \(X>1\) write NEWLINE\[NEWLINET(k,s;X)= \sum_{1\leq m_1,\dots,m_s\leq X}d(m^k_1+\cdots+ m^k_s).\tag{\(*\)}NEWLINE\]NEWLINE When \(k= 2\), \(s\in\{2,3,4\}\), asymptotic formulae have been established for \((*)\) by various authors. The earliest such paper [Dokl. Akad. Nauk Tadzh. SSR 28, 371--375 (1985; Zbl 0586.10023)] for the case \(k=2\), \(s=2\) is due to \textit{N. Gafurov} who published another paper on this topic in 1993. The first two main terms derived for the estimate of \((*)\) for the cases \(k=2\), \(s\in\{2,3,4\}\), take the form \(AX^s\log X+ BX^s\) for certain constants \(A\), \(B\).NEWLINENEWLINE The aim of the current paper is to investigate \(T(k,s;X)\) in the cases \(k=2\), \(s\geq 3\) and \(k\geq 3\), \(s>\min(2^{k-1}, k^2+k-2)\). For these cases the constants \(A\), \(B\) above are given explicitly and an error term, as good as or better than that previously found in the cases \(k=2\), \(s= 2,3,4\), is derived. To prove her results the author uses the classical circle method when \(k\geq 3\) and the Hardy-Littlewood-Kloosterman circle method when \(k=2\). The proof draws on lemmas previously published by other authors. The paper is clearly presented and well written.