Mertens' Third Theorem

Theorem

$\ds \lim_{x \mathop \to \infty} \ln x \prod_{\substack {p \mathop \le x \\ \text {$p$ prime} } } \paren {1 - \dfrac 1 p} = e^{-\gamma}$

where $\gamma$ denotes the Euler-Mascheroni constant.


Proof


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Source of Name

This entry was named for Franz Mertens.


Sources

  • 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,56145 94835 66885 \ldots$
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