Asymptotic of some sums

Abstract

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

9 pages