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]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <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]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 14:40, 24 January 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
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: █
