Theorem of Mertens

One usually indicates under the name of first theorem of Mertens the following estimate, where, by convention, a nap (or a produced) subscripted by $p$ indicates a sum (or a product) carrying only on the prime numbers. For all real $x\; \backslash \; geq\; 2$, one a:

$\backslash \; sum\_\; \{p\; \backslash \; Leq\; X\}\; \backslash \; frac\; \{\backslash \; ln\; p\}\; \{p\}\; =\; \backslash \; ln\; X\; +\; O\; (1).$

The demonstration uses the Formule of Legendre on the p-adic valuations of $n!$.

The Second theorem of Mertens , as called formula of Mertens , stipulates as, for any reality $x\; \backslash \; geq\; 2$, one a:

$\backslash \; prod\_\; \{p\; \backslash \; Leq\; X\}\; \backslash \; left\; (1\; -\; \backslash \; frac\; \{1\}\; \{p\}\; \backslash \; right)\; =\; \backslash \; frac\; \{e^\; \{-\; \backslash \; gamma\}\}\; \{\backslash \; ln\; X\}\; \backslash \; left\; (1\; +\; O\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{\backslash \; ln\; X\}\; \backslash \; right)\; \backslash \; right),$

where $\backslash \; gamma\; \backslash \; approx\; 0,577\; 215.664\dots$ is the Constante of Euler-Mascheroni.

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