Problems similar to the additive divisor problem

Abstract

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants$A_\varepsilon$,α>0 such that$f(n) \leqslant A_\varepsilon n^\varepsilon$ and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

$$\sum\limits_{n \leqslant x} {f(n)} \tau (n - 1) = C(f)\sum\limits_{n \leqslant x} {f(n)lnx(1 + 0(1))}$$

. Hereτ(n) is the number of divisors ofn andC(ƒ) is a constant.