On the mean square formula of the error term for a class of arithmetical functions (Q1807621)

The author investigates mean square formulas for the error term of a class of arithmetic functions satisfying certain general conditions. This class, introduced by \textit{K. Chandrasekharan} and \textit{R. Narasimhan} [see their works in Math. Ann. 152, 30-64 (1963; Zbl 0116.27001) and Acta Math. 112, 41-67 (1964; Zbl 0128.26302)], was extensively studied by several authors. It includes, for example, the functions \(d(n) = \sum_{k\ell=n}1 (k,\ell\in\mathbb N)\) and \(r(n) = \sum_ {n=a^2+b^2}1 (a,b\in\mathbb Z)\). If \(\phi(s) = \sum_{n=1}^\infty a_n\lambda_n^{-s}\) is the corresponding (generating) Dirichlet series, then the error term in question is, for suitable \(\rho\in\mathbb R\), \[ E_\rho(x) := {1\over\Gamma(\rho+1)}{\sum_{\lambda_n\leq x}}' a_n(x-\lambda_n) - M_\rho(x),\quad M_\rho(x) = \sum_\xi c_\xi x^\xi\log^{{r_\xi}-1}x, \] where \( ' \) means that the last term in the sum is halved if \(x = \lambda_n\), and the main term \(M_\rho(x)\) arises from the residues of the poles of \(\phi(s)/s\). The author proves two interesting theorems, whose formulation is too technical to be given here in detail, by using the method of \textit{T. Meurman} [Acta Arith. 74, 351-364 (1996; Zbl 0848.11043)], who applied it to a particular divisor function. The second theorem provides estimates for the error term in the asymptotic formula for \(\int_{x/2}^x|E_0(y)|^2 \text{ d}y\), where the problem is to get the power of the logarithm in the error term as small as possible. The result gives the sharpest known bounds in some well-known cases (which include the functions \(d(n)\) and \(r(n)\)), while at the same time it furnishes new results in several interesting cases.