Meissel-Mertens constant

The Meissel-Mertens constant (after Ernst Meissel and Franz Mertens ) is a mathematical constant . Similar to the sum of the reciprocal natural numbers ( harmonic series ), the sum of the reciprocal prime numbers also grows indefinitely (here describes the set of all prime numbers). I.e. both sums are arbitrarily large for an increasing number of members n . The exact asymptotic growth is described by the two limit values: ${\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}}$ ${\displaystyle \sum _{p\in \mathbb {P} }^{n}{\frac {1}{p}}}$ ${\displaystyle \mathbb {P} }$

${\displaystyle \gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n\right),\qquad M:=\lim _{n\to \infty }\left(\sum _{{p\in \mathbb {P} } \atop {p\leq n}}{\frac {1}{p}}-\ln(\ln n)\right)}$

Here is the Euler's constant and the Meissel-Mertens constant. The sum of all reciprocal prime numbers between 2 and n therefore grows asymptotically like the nested logarithm . It occurs mainly in number theory and function theory . There are numerous relationships with other mathematical constants and series . For example: ${\displaystyle \gamma }$ ${\displaystyle M}$ ${\displaystyle \ln(\ln n)\ }$

${\displaystyle M=\gamma +\sum _{p\in \mathbb {P} }\left[\ln \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right]}$

${\displaystyle M=\gamma +\sum _{k=2}^{\infty }{\frac {\mu (k)}{k}}\ln {\bigg (}\zeta (k){\bigg )}}$

Here is the Möbius function and the Riemann zeta function . The numerical value of the Meissel-Mertens constant is ${\displaystyle \mu (n)}$ ${\displaystyle \zeta (n)}$

${\displaystyle M=0{,}26149\;72128\;47642\;78375\;54268\;38608\;69585\;90515\;66648\;26119\ldots }$(Follow A077761 in OEIS )

literature

• Franz Mertens : A Contribution to Analytical Number Theory. In: Journal for pure and applied mathematics , 78, 1874, pp. 46–62 ( GDZ )

Web links

• Eric W. Weisstein : Mertens Constant . In: MathWorld (English).
• Episode A096167 in OEIS (Engel development of M )