Mertens' theorems

In mathematics, Mertens' theorems are three results in number theory related to the density of prime numbers and one result in analysis, and proved by Franz Mertens.

In the following, let $p < n$ mean all primes not exceeding $n$ .

Mertens' 1st theorem

$\lim_{n\to\infty}\left( \ln n-\sum_{p<n}\frac{\ln p}p \right)=1.3325822757\ldots$

Mertens' 2nd theorem

$\lim_{n\to\infty}\left(-\ln\ln n+\sum_{p<n}\frac1p\right)=0.2614972128\ldots,$

Mertens' 3rd theorem, 1874:

$\lim_{n\to\infty}\ln n\prod_{p<n}\left(1-\frac1p\right)=e^{-\gamma},$

Mertens's theorem can also refer to the result that if a real or complex infinite series

$\sum_{n=1}^\infty a_n$

converges to $A$ and another

$\sum_{n=1}^\infty b_n$

[edit] Further reading

Yaglom and Yaglom Challenging mathematical problems with elementary solutions Vol 2, problems 171, 173, 174