On a divisor problem related to the Epstein zeta-function. IV (Q2896951)

The authors continue their previous studies of divisor problems related to the Epstein zeta-function [Bull. Lond. Math. Soc. 42, No. 2, 267--274 (2010; Zbl 1215.11092), J. Number Theory 131, No. 9, 1734--1742 (2011; Zbl 1283.11139), Q. J. Math. 63, No. 4, 953--963 (2012; Zbl 1320.11094)]. For the asymptotic behavior of the error term NEWLINE\[NEWLINE\Delta_k^*(x,Q):= \sum_{n\leq x} r_k(n,Q) -\text{Res}_{s=\ell/2}(Z_Q(s)^k x^s s^{-1}NEWLINE\]NEWLINE \textit{A. Sankaranarayanan} [Arch. Math. 65, No. 4, 303--309 (1995; Zbl 0839.11042)] showed by complex integration that for \(k\geq 2\) and \(\ell\geq 3\) NEWLINE\[NEWLINE\Delta_k^*(x,Q)\ll x^{\ell/2-1/k+\varepsilon}.\tag{1}NEWLINE\]NEWLINE This was improved by the authors in Part II using a simple convolution argument to NEWLINE\[NEWLINE\Delta_k^*(x,Q)\ll x^{\ell/2-1+\theta_k+\varepsilon}\quad (x\geq 2)NEWLINE\]NEWLINE for \(k=2,3\) where \(\theta_k\) is the exponent in the classical divisor problem. Moreover, an \(\Omega\)-result for \(k=2,3\) and a mean value theorem for \(\Delta_2^*(x,Q)\) have been also established in Part II and III. NEWLINENEWLINEIn the present paper the authors refine Sankaranarayanan's result (1) for general definite quadratic forms \(Q\) when \(k\geq 3\). They make use of the analytic continuation of \(L_Q(s)\), subconvexity bounds for \(L_Q(s)\) similar to \(\zeta(s)\), and complex integration.