An extension of the Siegel-Walfisz theorem

TL;DR

This paper generalizes the Siegel-Walfisz theorem to a broader class of integer sequences defined by prime factor size constraints, including smooth numbers, almost primes, and practical numbers.

Contribution

It extends the classical theorem to new families of integers characterized by prime factor size restrictions.

Findings

Extended the Siegel-Walfisz theorem to these sequences.

Provided bounds and asymptotic estimates for the distribution of these integers.

Unified treatment of various special number classes under the theorem's framework.

Abstract

We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family includes smooth numbers, almost primes and practical numbers.