Meissel–Mertens Constant on:
[Wikipedia]
[Google]
[Amazon]
The Meissel–Mertens constant (named after Ernst Meissel and 
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
Franz Mertens
Franz Mertens (20 March 1840 – 5 March 1927) (also known as Franciszek Mertens) was a German-Polish mathematician. He was born in Schroda in the Grand Duchy of Posen, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in Vienna, Au ...
), also referred to as the Mertens constant, Kronecker's constant (after Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker
as having said, ...
), Hadamard–de la Vallée-Poussin constant (after Jacques 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 tea ...
and Charles Jean de la Vallée-Poussin
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning wa ...
), or the prime reciprocal constant, is a mathematical constant
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...
in number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, defined as the limiting 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 a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
of the natural logarithm:
:
Here γ is the Euler–Mascheroni constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
, 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 analysis, 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 p ...
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 and technology company focusing on online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, consumer electronics, and artificial ...
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. It was founded in ...
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 ...
), $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-cent ...
* 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
* * * * {{DEFAULTSORT:Meissel-Mertens constant Mathematical constants