Mertens' theorems

In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.

In the following, let ${\displaystyle p\leq n}$ mean all primes not exceeding n.

Mertens' first theorem is that

does not exceed 2 in absolute value for any ${\displaystyle n\geq 2}$. (A083343)

Mertens' second theorem is

where M is the Meissel–Mertens constant (A077761). More precisely, Mertens proves that the expression under the limit does not in absolute value exceed

for any ${\displaystyle n\geq 2}$.

The main step in the proof of Mertens' second theorem is