Estimation of multiple sums of arithmetical functions (Q2756170)

Let \(f(d_1,\dots, d_m)\) be an arithmetic function which is multiplicative in the variables \(d_1,\dots, d_m\), and suppose that the Dirichlet series NEWLINE\[NEWLINEF(s_1,\dots, s_m)= \sum^\infty_{d_1,\dots, d_m=1} \frac{f(d_1,\dots, d_m)} {d_1^{s_1}\dots d_m^{s_m}}NEWLINE\]NEWLINE is suitably well behaved. This paper shows how one may use a multidimensional version of Perron's formula to estimate NEWLINE\[NEWLINE\sum_{d_1\leq N_1}\dots \sum_{d_m\leq N_m} f(d_1,\dots, d_m).NEWLINE\]NEWLINE If the \(N_i\) are all powers of a parameter \(X\), an asymptotic formula with a relative saving \(O(X^{-\delta})\) is obtained. The calculation of the main term, arising from residues of \(F(s_1,\dots, s_m)\), is of particular interest.