Mertens' prime product formula, dissected

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.