Mertens\' theorems - Citizendia

In mathematics, Mertens' theorems are three 1874 results in number theory related to the density of prime numbers and one result in analysis, and proved by Franz Mertens. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Year 1874 ( MDCCCLXXIV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Common Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Analysis has its beginnings in the rigorous formulation of Calculus. Franz Mertens ( March 20, 1840 - March 5, 1927) was a German Mathematician.

In the following, let $p < n$ mean all primes not exceeding $n$ .

Mertens' 1st theorem:

$\ln n - \sum_{p < n} \frac{\ln p}{p} = O(1) \quad \hbox{as}\ n\to\infty,$

see Big O notation. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments

Mertens' 2nd theorem:

$\lim_{n\to\infty}\left(-\ln\ln n+\sum_{p<n}\frac1p\right)=0.2614972128\ldots,$

see 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

Mertens' 3rd theorem:

$\lim_{n\to\infty}\ln n\prod_{p<n}\left(1-\frac1p\right)=e^{-\gamma},$

where γ is the Euler-Mascheroni constant. The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually

Mertens' theorem can also refer to the result that if a real or complex infinite series

$\sum_{n=1}^\infty a_n$

converges to $A$ and another

$\sum_{n=1}^\infty b_n$

converges absolutely to $B$ then their Cauchy product converges to $A B$ . In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the In Mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two Sequences a_n b_n is the discrete Convolution In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or

External links

Eric W. Weisstein, Mertens Constant at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Sondow, Jonathan and Weisstein, Eric W. , Mertens Theorem at MathWorld. MathWorld is an online Mathematics reference work created and largely written by Eric W
Eric W. Weisstein, Mertens' Second Theorem at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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