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Meissel–Mertens constant

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamardde la Vallée-Poussin constant or prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series of the primes and the natural logarithm of the natural logarithm, i.e.

${\displaystyle M:=\lim _{k\to \infty }\left\{\left(\sum _{i=1}^{k}{\frac {1}{p_{i}}}\right)-\log \log p_{k}\right\}=\lim _{n\rightarrow \infty }\left\{\left(\sum _{p\leq n}{\frac {1}{p}}\right)-\log \log n\right\},\,}$

and

${\displaystyle M=\gamma \,+\lim _{k\to \infty }\left\{\left(\sum _{i=1}^{k}{\frac {1}{p_{i}}}\right)-\log \prod _{i=1}^{k}\left({\frac {1}{1-{\frac {1}{p_{i}}}}}\right)\right\}=\gamma \,+\lim _{k\to \infty }\left\{\sum _{i=1}^{k}\left[{\frac {1}{p_{i}}}-\log \left({\frac {1}{1-{\frac {1}{p_{i}}}}}\right)\right]\right\}=\gamma \,+\sum _{\stackrel {p}{p{\rm {~prime}}}}\left[{\frac {1}{p}}+\log \left(1-{\frac {1}{p}}\right)\right],\,}$

where ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$-th prime and ${\displaystyle \scriptstyle \gamma \,}$ is the Euler–Mascheroni constant, which is the analogous constant involving the harmonic series (which is over all positive integers).

Mertens' second theorem asserts that the limit exists. (Is it known whether the Meissel–Mertens constant is rational or irrational.)

The fact that there are two logarithms (log of a log) in the limit for the Meissel–Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler–Mascheroni constant.

Decimal expansion of the Meissel–Mertens constant

The decimal expansion of the Meissel–Mertens constant ${\displaystyle \scriptstyle M\,}$ is

${\displaystyle M=0.2614972128476427837554268386086958590516\ldots ,\,}$

A077761 Decimal expansion of Mertens' constant, which is the limit of Sum{1/p(i), i=1..k } - log(log(p(k))) as k goes to infinity, where p(i) is the i-th prime number.

{2, 6, 1, 4, 9, 7, 2, 1, 2, 8, 4, 7, 6, 4, 2, 7, 8, 3, 7, 5, 5, 4, 2, 6, 8, 3, 8, 6, 0, 8, 6, 9, 5, 8, 5, 9, 0, 5, 1, 5, 6, 6, 6, 4, 8, 2, 6, 1, 1, 9, 9, 2, 0, 6, 1, 9, 2, 0, 6, 4, 2, 1, 3, 9, 2, 4, 9, ...}

Continued fraction for the Meissel–Mertens constant

The simple continued fraction for the Meissel–Mertens constant is

${\displaystyle M=0+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}\,}$

A230767 Continued fraction for the Meissel–Mertens constant.

{0, 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, ...}

Noncomposite reciprocal constant

The noncomposite reciprocal constant is given by

${\displaystyle {\overline {C}}:=\lim _{k\to \infty }\left\{\left(\sum _{i=1}^{k}{\frac {1}{q_{i}}}\right)-\log \log q_{k}\right\}=\lim _{n\to \infty }\left\{\left(\sum _{\stackrel {q\leq n}{q{\rm {~noncomposite}}}}{\frac {1}{q}}\right)-\log \log n\right\}=\gamma -C=M+1,\,}$

where ${\displaystyle \scriptstyle q_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$-th [positive] noncomposite (one or prime) and ${\displaystyle \scriptstyle C\,}$ is the composite reciprocal constant.

Decimal expansion of the noncomposite reciprocal constant

The decimal expansion of the noncomposite reciprocal constant is

${\displaystyle {\overline {C}}=\gamma -C=M+1=1.2614972128476427837554268386086958590516....\,}$