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# Mathematics > Number Theory

# Title: Mertens' prime product formula, dissected

(Submitted on 9 Feb 2020 (v1), last revised 16 Mar 2021 (this version, v3))

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.

## Submission history

From: Jared Duker Lichtman [view email]**[v1]**Sun, 9 Feb 2020 13:41:19 GMT (6kb)

**[v2]**Sat, 27 Jun 2020 21:51:17 GMT (8kb)

**[v3]**Tue, 16 Mar 2021 21:59:24 GMT (11kb)

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