On an approximation by Vaughan in restricted sets of arithmetic progressions

TL;DR

This paper studies Vaughan's approximation for prime counts in specific subsets of arithmetic progressions with additional congruence conditions, providing new asymptotic formulas for these restricted sets.

Contribution

It extends Vaughan's approximation to subsets of arithmetic progressions satisfying extra congruence conditions, offering new asymptotic results.

Findings

Derived asymptotic formulas for prime counts in restricted arithmetic progressions.

Showed Vaughan's approximation remains effective in subsets with additional congruence constraints.

Enhanced understanding of prime distribution in specialized arithmetic progression subsets.

Abstract

We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of arithmetic progressions that satisfy additional congruence conditions and provide asymptotic approximations.