Asymptotic of summation functions

TL;DR

This paper investigates the asymptotic behavior of summation functions involving natural and prime arguments, deriving a general formula based on prime distribution laws and establishing necessary and sufficient conditions for its validity.

Contribution

It introduces a general formula for the asymptotics of sums over prime arguments and proves the conditions under which it holds.

Findings

Derived a formula linking prime sums to natural sums with logarithmic weights

Established necessary and sufficient conditions for the formula's validity

Analyzed asymptotic behavior of summation functions involving primes

Abstract

We will study the asymptotic behavior of summation functions of a natural argument, including the asymptotic behavior of summation functions of a prime argument in the paper. A general formula is obtained for determining the asymptotic behavior of the sums of functions of a prime argument based on the asymptotic law of primes. We will show, that under certain conditions: $\sum_{p \leq n} {f(p)}= \sum_{k=2}^n {\frac {f(k)}{\log(k)}(1+o(1))}$, where $p$ is a prime number. In the paper, the necessary and sufficient conditions for the fulfillment of this formula are proved.