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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Franz Mertens.

Sources

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