Mertens' prime product formula, dissected

TL;DR

This paper provides a detailed dissection of Mertens' prime product formula by deriving series expansions for numbers with a fixed number of prime factors, using elementary combinatorial methods.

Contribution

It introduces new formulas for series over numbers with a fixed number of prime factors, dissecting Mertens' original estimate with elementary combinatorial techniques.

Findings

Derived formulas for series over numbers with fixed prime factors

Dissected Mertens' prime product formula into component series

Used elementary combinatorial methods for proofs

Abstract

In 1874, Mertens famously proved an asymptotic formula for the product $p/(p-1)$ over all primes $p$ up to $x$. On the other hand, one may expand Mertens' prime product into series over numbers $n$ with only small prime factors. It is natural to restrict such series to numbers $n$ with a fixed number $k$ of prime factors. In this article, we obtain formulae for these series for each $k$, which together dissect Mertens' original estimate. The proof is by elementary methods of a combinatorial flavor.