# 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)}.

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

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Source: https://tomesphere.com/paper/1901.06160