On the mean values of certain arithmetic functions
Let \(d_k (n)\) be the number of representations of \(n\) as a product of \(k\) natural numbers. Then \[ \sum_{n\leq x} d_k (n)= x P_k (\log x)+ \Delta_k (x), \] where \(P_k\) is a polynomial of degree \(k-1\). \textit{A. A. Karatsuba} [Math. USSR, Izv. 6 (1972), 467--475 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, No. 3, 475--483 (1972; Zbl 0239.10025)]\ proved that there are two constants \(c>0\), \(c_1> 0\) such that the estimation \[ \Delta_k (x) \ll x^{1-c k^{-2/3}} (c_1\log x)^k \] holds for \(k= \log x\). In this paper similar estimates are proved for the mean values of the functions \(d_k (n^2), d_{k_1} (n) d_{k_2} (n) \dots d_{k_r} (n)\).
scientific article; zbMATH DE number 796786
scientific article