Difference between revisions of "Meissel-Mertens constant"

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=Properties=
 
=Properties=
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[[Meissel-Mertens constant in terms of the Euler-Mascheroni constant]]
<strong>Theorem:</strong> 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]].
 
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<strong>Proof:</strong> █
 
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=See Also=
 
=See Also=

Latest revision as of 00:28, 20 August 2016

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

Meissel-Mertens constant in terms of the Euler-Mascheroni constant

See Also

Euler-Mascheroni constant