Asymptotic of some sums

Abstract

The paper compares the asymptotic of the expressions $\frac {1} {x} \sum\limits_{n \leq x} {f(n)}$ and $\sum\limits_{n \leq x} {\frac {f(n)} {n}}$, $\frac {1} {x} \sum\limits_{p \leq x} {f(p)}$ and $\sum\limits_{p \leq x} {\frac {f(p)} {p}}$. The asymptotic of sums $\sum\limits_{n \leq x} {\frac {f(n)} {n}}$ and $\sum\limits_{p \leq x} {\frac {f(p)} {p}}$ ($n,p$ - respectively, positive and prime numbers) are determined if the asymptotic of sums are known, respectively: $\sum\limits_{n \leq x} {f(n)}$,$\sum\limits_{p \leq x} {f(p)}$.