Asymptotic of some sums

TL;DR

This paper investigates the asymptotic behavior of various sums involving a function f(n), comparing sums over natural numbers and primes, and establishing relations between their asymptotics.

Contribution

It provides new asymptotic formulas for sums of the form rac{f(n)}{n} and rac{f(p)}{p} based on known asymptotics of rac{f(n)} and rac{f(p)}.

Findings

Asymptotic formulas for rac{f(n)}{n} and rac{f(p)}{p} derived from known sums.

Relations established between sums over natural numbers and primes.

Conditions under which asymptotics of these sums are equivalent or related.

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: $\sum\limits_{n \leq x} {f(n)}$,$\sum\limits_{p \leq x} {f(p)}$.