Meissel-Mertens constant

The Meissel-Mertens constant (also known as Mertens' constant, Kronecker's constant, the Hadamard-de la Vallée-Poussin constant, or prime reciprocal constant) is $$M=\displaystyle\lim_{n \rightarrow \infty} \left( \displaystyle\sum_{p \leq n;p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p} - \log(\log(n)) \right).$$ Note that the sum $\displaystyle\sum_{p \leq n;p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p}$ diverges, so this definition resembles that of the Euler-Mascheroni constant.

Properties

Theorem: The Meissel-Mertens constant can be written as $$M=\gamma + \displaystyle\sum_{p \leq n;p \mathrm{\hspace{2pt} prime}} \left[ \log \left( 1 - \dfrac{1}{p} \right) + \dfrac{1}{p} \right],$$ where $\gamma$ denotes the Euler-Mascheroni constant and $\log$ denotes the logarithm.

Proof: █