Mertens' theorems

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

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

Mertens' 1st theorem:

$\ln n-\sum _{p\leq n}{\frac {\ln p}{p}}=O(1)\quad {\hbox{as}}\ n\to \infty ,$

where O(1) is Big O notation.

Mertens' 2nd theorem:

$\lim _{n\to \infty }\left(-\ln \ln n+\sum _{p\leq n}{\frac {1}{p}}\right)=M,$

where M is the Meissel–Mertens constant.

Mertens' 3rd theorem:

$\lim _{n\to \infty }\prod _{p\leq n}\left(1-{\frac {1}{p}}\right)\log n=e^{-\gamma },$

where γ is the Euler–Mascheroni constant.

In 2010 N. A. Carella proved that in Mertens' 2nd theorem the difference

$-\ln \ln n-M+\sum _{p\leq n}{\frac {1}{p}}$

changes sign infinitely often. Earlier Harold Diamond and János Pintz had proved that in Mertens' 3rd theorem the difference

$\ln n\prod _{p\leq n}\left(1-{\frac {1}{p}}\right)-e^{-\gamma }$

changes sign infinitely often. These results are analogous to Littlewood's famous theorem that the difference π(x) − li(x) changes sign infinitely often (see Skewes number). No analog of the Skewes number (an upper bound on the first natural number x for which π(x) > li(x)) is known in the case of Mertens' 2nd and 3rd theorems.

In summability theory, Mertens' theorem states that if a real or complex infinite series

$\sum _{n=1}^{\infty }a_{n}$

converges to A and another

$\sum _{n=1}^{\infty }b_{n}$

converges absolutely to B then their Cauchy product converges to AB.

Further reading

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

External links

• Weisstein, Eric W. "Mertens Constant". MathWorld.
• Sondow, Jonathan and Weisstein, Eric W. "Mertens Theorem". MathWorld.{{cite web}}: CS1 maint: multiple names: authors list (link)
• Weisstein, Eric W. "Mertens' Second Theorem". MathWorld.