Asymptotic of summation functions

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: ${\sum }_{p\le n}f(p)={\sum }_{k=2}^n\frac{f(k)}{\log (k)}(1+o(1))$, where $p$ is a prime number. In the paper, the necessary and sufficient conditions for the fulfillment of this formula are proved.

15 pages