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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 the prime reciprocal constant, is a 
The value of ''M'' is approximately
:''M'' ≈ 0.2614972128476427837554268386086958590516... .
Mertens' second theorem establishes that the limit exists.
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.
''On the remainder in a series of Mertens''
(postscript file) {{DEFAULTSORT:Meissel-Mertens constant Mathematical constants
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 the prime reciprocal constant, is a mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of the natural logarithm:
:
Here γ is the Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
, which has an analogous definition involving a sum over all integers (not just the primes).
The value of ''M'' is approximately
:''M'' ≈ 0.2614972128476427837554268386086958590516... .
Mertens' second theorem establishes that the limit exists.
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.
In popular culture
The Meissel-Mertens constant was used byGoogle
Google LLC () is an American Multinational corporation, multinational technology company focusing on Search Engine, search engine technology, online advertising, cloud computing, software, computer software, quantum computing, e-commerce, ar ...
when bidding in the Nortel
Nortel Networks Corporation (Nortel), formerly Northern Telecom Limited, was a Canadian Multinational corporation, multinational telecommunications and data networking equipment manufacturer headquartered in Ottawa, Ontario, Canada. It was foun ...
patent auction. Google posted three bids based on mathematical numbers: $1,902,160,540 ( Brun's constant), $2,614,972,128 (Meissel–Mertens constant), and $3.14159 billion ( π).
See also
*Divergence of the sum of the reciprocals of the primes
The sum of the reciprocals of all prime numbers diverges; that is:
\sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty
This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-centur ...
* Prime zeta function In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for \Re(s) > 1:
:P(s)=\sum_ \frac=\frac+\frac+\frac+\frac+\frac+\cdots.
Properties
...
References
External links
*''On the remainder in a series of Mertens''
(postscript file) {{DEFAULTSORT:Meissel-Mertens constant Mathematical constants