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.

arXiv commentInterprets Corollary 1.5 in terms of "friable regularity," under extended definition from convergent series to partial sums of arbitrary sequences. Incorporates referee comments

Cited by in corpus (1)

• Translated sums of primitive sets