Large values of error terms of a class of arithmetical functions (Q2781426)

The aim of the paper is to establish \(\Omega_\pm\)-estimations of the error terms in the asymptotic representations of the mean values of arithmetic functions. A class of functions is considered whose Dirichlet series satisfy functional equations with multiple gamma factors. This class contains many classical examples, such as divisor functions and enumerating functions of representations of an integer by quadratic forms. The main result improves Theorem 1 of \textit{A. Ivić} [Acta Arith. 56, 135-159 (1990; Zbl 0659.10053)]. NEWLINENEWLINENEWLINEWe mention a special case. Let \(r(Q,n)\) denote the number of representations of an integer \(n\) by a positive definite ternary quadratic form and NEWLINE\[NEWLINEP_Q(x)= \sum_{n\leq x}r(Q,n)-|\det Q|^{-1/2} \frac{(2\pi)^{3/2}} {\Gamma(5/2)} x^{3/2}.NEWLINE\]NEWLINE Then NEWLINE\[NEWLINEP_Q(x)= \Omega_\pm (\sqrt{x\log x}).NEWLINE\]NEWLINE This improves a result of \textit{S. D. Adhikari} and \textit{Y.-F. S. Pétermann} [Acta Arith. 59, 329-338 (1991; Zbl 0705.11056)].