Higher Mertens constants for almost primes

TL;DR

This paper provides elementary proofs for precise asymptotics of the reciprocal sum of k-almost primes, explores higher Mertens constants, and offers refined estimates for semiprimes, advancing understanding of prime factorization distributions.

Contribution

It introduces elementary proofs for asymptotics of k-almost primes and investigates higher Mertens constants, including explicit estimates for semiprimes.

Findings

Asymptotic formulas for reciprocal sums of k-almost primes

Identification of higher Mertens constants and their limiting behavior

Refined estimates and conjecture for semiprimes

Abstract

For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match the strength of those of classical analytic methods. We also study the limiting behavior of the constants appearing in these estimates, which may be viewed as higher analogues of the Mertens constant $\beta=0.2614...$ Further, in the case $k=2$ of semiprimes we give yet finer-scale and explicit estimates, as well as a conjecture.