# Meissel–Mertens constant

﻿
Meissel–Mertens constant

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamardde la Vallée-Poussin constant or prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:

$M = \lim_{n \rightarrow \infty } \left( \sum_{p \leq n} \frac{1}{p} - \ln(\ln(n)) \right)=\gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right].$

Here γ is the famous Euler–Mascheroni constant, which has a similar definition involving a sum over all integers (not just the primes).

The value of M is approximately

M ≈ 0.2614972128476427837554268386086958590516... (sequence A077761 in OEIS).

Mertens' 2nd theorem says 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.

The number was used as a bid in the Nortel patent auction. The bid posted by Google was one of three that were based on mathematical numbers.[1]

## References

1. ^ Reuters (July 5, 2011). "Google's strange bids for Nortel patents". FinancialPost.com. Retrieved 2011-08-16.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Meissel-Mertens constant — The Meissel Mertens constant, also referred to as Mertens constant, Kronecker s constant, Hadamard de la Vallée Poussin constant or prime reciprocal constant, is a mathematical constant, used mainly in number theory, and is defined as the… …   Wikipedia

• Meissel-Mertens-Konstante — Die Meissel Mertens Konstante (nach Ernst Meissel (1826–1895) und Franz Mertens) ist eine mathematische Konstante. Ähnlich wie die Summe der reziproken natürlichen Zahlen (harmonischen Reihe) wächst auch die Summe der reziproken Primzahlen… …   Deutsch Wikipedia

• Constante de Meissel-Mertens — En mathématiques, la constante de Meissel (de) Mertens (également nommée constante de Mertens, constante de Kronecker, constante de Hadamard de La Vallée Poussin ou constante des inverses des nombres premiers) est utilisée principalement en… …   Wikipédia en Français

• Constante De Meissel-Mertens — En mathématiques, la constante de Meissel Mertens (également nommée dans la littérature mathématique constante de Mertens, constante de Kronecker, constante d Hadamard de la Vallée Poussin ou constante des inverses des nombres premiers) est une… …   Wikipédia en Français

• Mertens — may refer to: In places: Mertens, Texas, a US town People with the surname Mertens: Dries Mertens, Belgian footballer Franz Mertens, German mathematician Franz Carl Mertens, German botanist Jan Mertens, Dutch politician Jan Mertens the Younger,… …   Wikipedia

• Mertens' theorems — For Mertens theorem on convergence of Cauchy products of series, see Cauchy product#Convergence and Mertens.27 theorem. In number theory, Mertens theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens… …   Wikipedia

• Euler–Mascheroni constant — Euler s constant redirects here. For the base of the natural logarithm, e ≈ 2.718..., see e (mathematical constant). The area of the blue region is equal to the Euler–Mascheroni constant. List of numbers – Irrational and suspected irrational… …   Wikipedia

• Franz Mertens — (March 20, 1840 March 5, 1927) was a German mathematician. He was born in Środa in the Grand Duchy of Poznań, Kingdom of Prussia (now Środa Wielkopolska, Poland) and died in Vienna, Austria.The Mertens function is the sum function for the Möbius… …   Wikipedia

• Mathematical constant — A mathematical constant is a special number, usually a real number, that is significantly interesting in some way .[1] Constants arise in many different areas of mathematics, with constants such as e and π occurring in such diverse contexts as… …   Wikipedia

• Brun's constant — In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called Brun s constant for twin primes and usually denoted by B 2 OEIS|id=A065421::B… …   Wikipedia