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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, Hadamard–de 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.
and
where is the -th prime and 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.
Contents
Decimal expansion of the Meissel–Mertens constant
The decimal expansion of the Meissel–Mertens constant is
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
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
where is the -th [positive] noncomposite (one or prime) and is the composite reciprocal constant.
Decimal expansion of the noncomposite reciprocal constant
The decimal expansion of the noncomposite reciprocal constant is
See also
External links
- Weisstein, Eric W., Mertens Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/MertensConstant.html]
- On the remainder in a series of Mertens (pdf file)
- Divergence of the sum of the reciprocals of the primes—Wikipedia.org.