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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_ \left( \left( \sum_^n \frac \right) - \ln (n) \right)=\int_1^\infty\left(-\right)\,dx]
Its approximate value is γ ≈ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335.

Contents

History

The constant was first defined by Swiss mathematician Leonhard Euler in a paper De Progressionibus harmonicus observationes published in 1735. Euler used the notation C for the constant, and 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.

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).

Properties

The constant is given by several integrals:

[\gamma = - \int_0^\infty \ln x }\,dx ]
:[ = - \int_0^1 \right ) }\,dx ]
:[ = \int_0^\infty }-\frac \right )e^ }\,dx ]
:[ = \int_0^\infty \left ( \frac-e^ \right ) }\,dx. ]
Other integrals that include [ \gamma ] are:

[ \int_0^\infty \ln(x) }\,dx = -1/4(\gamma+2 \ln2) \sqrt ]
[ \int_0^\infty (\ln(x))^2 }\,dx = \gamma^2 +1/6 \pi^2 .]
One can express [ \gamma ] also as a double integral (Sondow 2003, 2005) with equivalent series:

[ \gamma = \int_^\int_^ \frac \, dx\,dy = \sum_^\infty \left ( \frac-\ln \left ( \frac \right ) \right ). ]
An interesting comparison by J. Sondow (2005) is the double integral and alternating series

[ \ln \left ( \frac \right ) = \int_^\int_^ \frac \, dx\,dy = \sum_^\infty (-1)^ \left ( \frac-\ln \left ( \frac \right ) \right ). ]
It shows that [\ln \left ( \frac \right )] may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series (see Sondow 2005 #2)

[ \sum_^\infty \frac = \gamma ]
[ \sum_^\infty \frac = \ln \left ( \frac \right ) ]
where [ N_1(n) ] and [ N_0(n) ] are the number of 1's and 0's, respectively, in the base 2 expansion of [ n ].

The series for [ \gamma ] is equivalent to Vacca's interesting 1910 sum

[ \gamma = \sum_^\infty (-1)^m \frac ]
where [ \log_2 ] is the logarithm to the base 2 and [ \left \lfloor \, \right \rfloor ] is the floor function.

Vacca's series may be obtained by manipulation of Catalan's 1875 integral (see Sondow and Zudilin)

[ \gamma = \int_0^1 \frac \sum_^\infty x^ \, dx. ]

Relations to special functions

[ \gamma ] can also be expressed as an infinite sum with terms involving the values of the Riemann zeta function at positive integers:

[\gamma = \sum_^ \frac ]
[= \ln \left ( \frac \right ) + \sum_^ \frac \zeta(m+1)}. ]
Other Zeta-related series include

[ \gamma = \frac- \ln 2 - \sum_^\infty (-1)^m\,\frac [zeta(m)-1] ]
:[ = \lim_ \left [ frac - ln,n + sum_^n left ( frac - frac right ) right ]. ]
:[ = \lim_ \left [ frac} sum_^infty frac} sum_^m frac - n, ln2+ O left ( frac} right ) right ] ]
The error term in last identity is a rapidly decreasing function of n. As a result, the formula is well-suited to efficiently computing the constant to high precision.

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

[ \gamma = \lim_ \left [ frac) Gamma(n+1), n^})} - frac right ]. ]
Other interesting limits equaling the Euler-Mascheroni constant are the antisymmetric limit (Sondow, 1998)

[ \gamma = \lim_ \sum_^\infty \left ( \frac-\frac \right ) = \lim_ \left ( \zeta(s) - \frac \right ) ]
and

[ \gamma = \lim_ \left [ x - Gamma left ( frac right ) right ] ]
:[ = \lim_ \frac\, \sum_^n \left ( \left \lceil \frac \right \rceil - \frac \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_^n \frac - \ln(n) - \sum_^\infty \frac]
where [\zeta(s,k)] is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Hn. Expanding some of the terms in the Hurwitz zeta function gives:

[H_n = \ln n + \gamma + \frac - \frac + \frac - \varepsilon ], where [0 < \varepsilon < \frac .]
There is also the related limit:
[\gamma = \lim_ (H_ - \ln n).]
The constant can also be calculated as a derivative of Euler's Gamma function:
[\gamma = -\Gamma'(1).]

e to the power of γ

The constant eγ is also important in number theory. Occasionally, eγ is denoted [ y' ] It is expressed with the following limit, where pn is the n-th prime number:

[
e^\gamma = \lim_ \frac \prod_^n \frac ] which is a restatement of the third of Mertens' theorems. The numerical value of eγ is:
[e^\gamma =1.78107241799019798523650410310717954916964521430343\dots]
Other infinite products relating to [ e^ ] include

[ \frac}} = \prod_^\infty e^\,\left (1+\frac \right )^n ]
[ \frac} = \prod_^\infty e^\,\left (1+\frac \right )^n. ]
Both of these products result from the Barnes G-function.

We also have

[ e^ = \left ( \frac \right )^ \left (\frac \right )^ \left (\frac \right )^ \left (\frac \right )^ \cdots ]
where the nth factor is the (n+1)st root of
[\prod_^n (k+1)^}.]
This product is due to J. Ser (1926). It was rediscovered by J. Sondow (2003) using hypergeometric functions.

Appearances

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

References

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