Mertens' theorems

In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. "Mertens' theorem" may also refer to his theorem in analysis.

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

Mertens' 1st theorem:

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

Mertens' 2nd theorem:

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

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

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