Euler-Mascheroni constant

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The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm:

$\gamma = \lim_{n \rightarrow \infty } \left( \left( \sum_{k=1}^n \frac{1}{k} \right) - \ln(n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx$

Or more simply, it is the number that has this property,

Failed to parse (unknown error\fra): 1+\fra{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} \sim \ln n+\gamma

Its approximate value is γ ≈ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335

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History

The constant was first defined by Swiss mathematician Leonhard Euler in a paper De Progressionibus harmonicus observationes published in 1735.

Euler discovered that the harmonic series,

Failed to parse (unknown error\fra): 1+\fra{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}

can be approximated with the natural logarithm of n plus a certain constant, which he denoted C or

Failed to parse (unknown error\fra): 1+\fra{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} \sim \ln n+C

He came about this idea by noting that the area of the unit hyperbola, $f(x)=\frac{1}{x}$ , from 1 to n is $ln n$ , and that the area under the curve can also be approximated by the harmonic series, leading him to believe that the harmonic series can also be approximated using the natural logarithm.

Euler initially calculated its value to 6 decimal places. In 1761 he extended this calculation, publishing a value to 16 decimal places. In 1790 Italian mathematician Lorenzo Mascheroni introduced the notation γ for the constant, and attempted to extend Euler's calculation still further, to 32 decimal places, although subsequent calculations showed that he had made an error in the 20th decimal place.

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Properties

The constant is given by several integrals:

$\gamma = - \int_0^\infty { e^{-x} \ln(x) }\,dx$

$= - \int_0^1 { \ln\ln\left (\frac{1}{x}\right ) }\,dx$

$= \int_0^\infty {\left (\frac{1}{1-e^{-x}}-\frac{1}{x} \right )e^{-x} }\,dx$

$= \int_0^\infty { \frac{1}{x} \left ( \frac{1}{1+x}-e^{-x} \right ) }\,dx.$

Other integrals that include $γ$ are:

$\int_0^\infty { e^{-x^2} \ln(x) }\,dx = -1/4(\gamma+2 \ln2) \sqrt{\pi}$

$\int_0^\infty { e^{-x} (\ln(x))^2 }\,dx = \gamma^2 +1/6 \pi^2 .$

One can express $γ$ as a double integral also:

$\gamma = \int_{0}^{1}\int_{0}^{1} \frac{x-1}{(1-x\,y)\ln(x\,y)} \, dx\,dy$

An interesting comparison by J. Sondow (2005) is the double integral

$\ln \left ( \frac{4}{\pi} \right ) = \int_{0}^{1}\int_{0}^{1} \frac{x-1}{(1+x\,y)\ln(x\,y)} \, dx\,dy.$

It shows that $\ln \left ( \frac{4}{\pi} \right )$ may be thought of an "alternating Euler constant".

$γ$ can also be expressed as an infinite sum with terms involving the values of the Riemann zeta function at positive integers:

$\gamma = \sum_{m=2}^{\infty} \frac{(-1)^m\zeta(m)}{m}$

$= \ln \left ( \frac{4}{\pi} \right ) + \sum_{m=1}^{\infty} \frac{(-1)^{m-1} \zeta(m+1)}{2^m (m+1)}.$

In 1910, Vacca gave the interesting sum

$\gamma = \sum_{m=1}^\infty (-1)^m \frac{ \left \lfloor \log_2 m \right \rfloor}{n}$

where $log 2$ is the logarithm of base 2 and $\left \lfloor \, \right \rfloor$ is the floor function.

Vacca's series may be obtained by manipulation of Catalan's integral

$\gamma = \int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx.$

Other Zeta-related series include

$\gamma = \frac{3}{2}- \ln 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m} [\zeta(m)-1]$

$= \lim_{n \to \infty} \left [ \frac{2^n}{e^{2^n}} \sum_{m=0}^\infty \frac{2^{m \,n}}{(m+1)!} \sum_{t=0}^m \frac{1}{t+1} - n\, \ln2+ O \left ( \frac{1}{2^n\,e^{2^n}} \right ) \right ]$

$= \lim_{n \to \infty} \left [ \frac{2\,n-1}{2\,n} - \ln\,n + \sum_{k=2}^n \left ( \frac{1}{k} - \frac{\zeta(1-k)}{n^k} \right ) \right ].$

A limit related to the Beta function (in terms of Gamma functions) is

$\gamma = \lim_{n \to \infty} \left [ \frac{ \Gamma(\frac{1}{n}) \Gamma(n+1)\, n^{1+1/n}}{\Gamma(2+n+\frac{1}{n})} - \frac{n^2}{n+1} \right ].$

Two other interesting limits equaling the Euler-Macheroni constant are the antisymmetric limit

$\gamma = \lim_{s \to \infty} \sum_{n=1}^\infty \left ( \frac{1}{n^s}-\frac{1}{s^n} \right )$

and

$\gamma = \lim_{x \to \infty} \left [ x - \Gamma \left ( \frac{1}{x} \right ) \right ]$

$= \lim_{n \to \infty} \frac{1}{n}\, \sum_{k=1}^{n=1} \left ( \left \lceil \frac{n}{k} \right \rceil - \frac{n}{k} \right ).$

Closely related to this is the rational zeta series expression. By peeling off the first few terms of the series above, one obtains an estimate for the classical series limit:

$\gamma = \sum_{k=1}^n \frac{1}{k} - \ln(n) - \sum_{m=2}^\infty \frac{\zeta (m,n+1)}{m}$

where $ζ(s, k)$ is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, H_n. Expanding some of the terms in the Hurwitz zeta function gives:

$H_n = \ln n + \gamma + \frac {1} {2n} - \frac {1} {12n^2} + \frac {1} {120n^4} - \varepsilon$ , where $0 < \varepsilon < \frac {1} {252n^6}.$

There is also the related limit: $\gamma = \lim_{n \to \infty} (H_{n-1} - \ln n).$

The constant can also be calculated as a derivative of Euler's Gamma function:

γ = - Γ'(1).

The constant e^γ is also important in number theory. Occasionally, e^γ is denoted $y'$ It is expressed with the following limit, where p_n is the n-th prime number:

$e^\gamma = \lim_{n \to \infty} \frac {1} {\ln p_n} \prod_{i=1}^n \frac {p_i} {p_i - 1}$

which is a restatement of the third of Mertens' theorems. The numerical value of e^γ is:

$e^\gamma =1.78107241799019798523650410310717954916964521430343\dots$

Other infinite products relating to $e γ$ include

$\frac{e^{1+\gamma /2}}{\sqrt{2\,\pi}} = \prod_{n=1}^\infty e^{-1+1/(2\,n)}\,\left (1+\frac{1}{n} \right )^n$

$\frac{e^{3+2\gamma}}{2\, \pi} = \prod_{n=1}^\infty e^{-2+2/n}\,\left (1+\frac{2}{n} \right )^n.$

Both of these products result from the Barnes G-function

$e^{\gamma} = \left ( \frac{2}{1} \right )^{1/2} \left (\frac{2^2}{1 \cdot 3} \right )^{1/3} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/4} \cdots$

It is due to J. Sondow using hypergeometric functions.

It is not known whether γ is a rational number or not. However, continued fraction analysis shows that if γ is rational, its denominator has more than 10²⁴²⁰⁸⁰ digits (Havil, page 97).

The Euler-Mascheroni constant appears, among other places, in:

an inequality for Euler's totient function
the growth rate of the divisor function
a product formula for the gamma function
calculations of the digamma function
calculation of the Meissel-Mertens constant
expressions involving the exponential integral
the first term of the Taylor series expansion for the Riemann zeta function, where it is the first of the Stieltjes constants
the third of Mertens' theorems.

Author: Sand-reckoner 07 February 2006 (other authored)

Retrieved from "http://midnightwiki.com/wiki/Euler-Mascheroni_constant"

Categories: Mathematics | Author:Sand-reckoner

Euler-Mascheroni constant

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