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The Meissel–Mertens constant (named after 
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
''On the remainder in a series of Mertens''
(postscript file) {{DEFAULTSORT:Meissel-Mertens constant Mathematical constants
The Meissel–Mertens constant (named after Ernst Meissel
Daniel Friedrich Ernst Meissel (31 July 1826, Eberswalde, Brandenburg Province – 11 March 1895, Kiel) was a German astronomer who contributed to various aspects of number theory.
See also
*Meissel–Lehmer algorithm
*Meissel–Mertens constant
...
and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
– 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, defined as the limiting
In electronics, a limiter is a circuit that allows signals below a specified input power or level to pass unaffected while attenuating (lowering) the peaks of stronger signals that exceed this threshold. Limiting is a type of dynamic range comp ...
difference between the harmonic series summed only over the primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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 l ...
, 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
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
and the limit of the Euler–Mascheroni constant.
In popular culture
The Meissel-Mertens constant was used byGoogle
Google LLC () is an American multinational technology company focusing on search engine technology, online advertising, cloud computing, computer software, quantum computing, e-commerce, artificial intelligence, and consumer electronics. ...
when bidding in the Nortel
Nortel Networks Corporation (Nortel), formerly Northern Telecom Limited, was a Canadian multinational telecommunications and data networking equipment manufacturer headquartered in Ottawa, Ontario, Canada. It was founded in Montreal, Quebec, ...
patent auction. Google posted three bids based on mathematical numbers: $1,902,160,540 (Brun's constant
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by V ...
), $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-century ...
* 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