Prime Gaps and Twin Primes in Arithmetic Sequences

Abstract

This paper investigates the distribution patterns of prime gaps and twin prime pairs within specific arithmetic sequences. Building upon classical results in analytic number theory, we examine how prime gaps behave differently in arithmetic progressions compared to the general prime sequence. We analyze twin prime occurrences across various residue classes and derive asymptotic estimates for gap distributions. Our findings demonstrate that arithmetic sequences with common difference d exhibit characteristic gap patterns influenced by local density variations. Through numerical analysis and theoretical examination, we establish bounds for the expected number of twin primes in arithmetic sequences of the form a + nd, where gcd(a, d) = 1. The results contribute to understanding the interplay between sieve methods, the Hardy-Littlewood conjecture, and modern bounded gap theorems in constrained prime sets.