Difference between revisions of "Meissel-Mertens constant"
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$$M=\displaystyle\lim_{n \rightarrow \infty} \left( \displaystyle\sum_{p \leq n;p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p} - \log(\log(n)) \right).$$ | $$M=\displaystyle\lim_{n \rightarrow \infty} \left( \displaystyle\sum_{p \leq n;p \mathrm{\hspace{2pt} prime}} \dfrac{1}{p} - \log(\log(n)) \right).$$ | ||
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]]. | 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]]. | ||
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| + | =Properties= | ||
| + | [[Meissel-Mertens constant in terms of the Euler-Mascheroni constant]] | ||
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| + | =See Also= | ||
| + | [[Euler-Mascheroni constant]] | ||
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| + | [[Category:SpecialFunction]] | ||
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
