Asymptotics for sums of functions of primes

TL;DR

This paper develops a general method to analyze the asymptotic behavior of sums involving functions of primes, refining remainder estimates and establishing conditions for their existence based on prime distribution.

Contribution

It introduces a new approach for determining asymptotics of sums of functions of primes, including conditions for their existence and improved remainder estimates.

Findings

Refined estimates for the remainder term in asymptotic expansions

Necessary and sufficient conditions for the existence of asymptotics

A general framework based on prime distribution

Abstract

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums of functions of primes. Also, the necessary and sufficient conditions for the existence of these asymptotics are proved in the paper.