Asymptotics of sums of functions of primes located on an arithmetic progression

TL;DR

This paper derives asymptotic formulas for sums of functions of primes in arithmetic and geometric progressions, linking prime distribution to sum behavior, and establishes conditions for their existence.

Contribution

It provides a general asymptotic estimate for sums of functions of primes on arithmetic progressions and explores asymptotics on geometric progressions with necessary and sufficient conditions.

Findings

Derived a general asymptotic formula for sums of functions of primes on arithmetic progressions

Established asymptotics for sums of functions of primes on geometric progressions

Proved conditions for the existence of these asymptotics

Abstract

We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression. Based on this distribution, a general formula was obtained for the asymptotic estimate of the sums of functions of primes, and also asymptotics was found for the sums of various functions of primes on a geometric progression. Several assertions about asymptotics of sums of functions of prime numbers on a geometric progression are proved. Necessary and sufficient conditions for the existence of these asymptotics are also proved.