List of numbers – Irrational and
suspected irrational numbers γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δ_{S} – α – e – π – δ 

Binary  0.100100111100010001... 
Decimal  0.5772156649015328606065... 
Hexadecimal  0.93C467E37DB0C7A4D1BE... 
Continued fraction  [0; 1, 1, 2, 1, 2, 1, 4, 3,
13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ]^{[1]}
(This continued fraction is not periodic. Shown in linear notation) 
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter γ (gamma).
It is defined as the limiting difference between the harmonic series and the natural logarithm:
Its numerical value to 50 decimal places is
γ should not to be confused with the base of the natural logarithm, e, which is sometimes called Euler's number.
Contents 
The constant first appeared in a 1735 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notation C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835.^{[2]}
The Euler–Mascheroni constant appears, among other places, in ('*' means that this entry contains an explicit equation):
For more information of this nature, see Gourdon and Sebah (2004).
The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Continued fraction analysis reveals that if γ is rational, its denominator must be greater than 10^{242080}.^{[3]} The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics. Also see Sondow (2003a).
For more equations of the sort shown below, see Gourdon and Sebah (2002).
γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:
This is equal to the limits:
Further limit results are (Krämer, 2005):
A limit related to the Beta function (expressed in terms of gamma functions) is
γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:
Other series related to the zeta function include:
The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is wellsuited for efficient computation of the constant to high precision.
Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit (Sondow, 1998)
and
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:
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:
γ equals the value of a number of definite integrals:
Definite integrals in which γ appears include:
One can express γ using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:
An interesting comparison by J. Sondow (2005) is the double integral and alternating series
It shows that may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series (see Sondow 2005 #2)
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.
We have also Catalan's 1875 integral (see Sondow and Zudilin)
Euler showed that the following infinite series approaches γ:
The series for γ is equivalent to series Nielsen found in 1897:
In 1910, Vacca found the closely related series:
where log_{2} is the logarithm to the base 2 and is the floor function.
In 1926 he found a second series:
From the Kummerexpansion of the gamma function we get:
γ equals the following asymptotic formulas (where H_{n} is the nth harmonic number.)
The third formula is also called the Ramanujan expansion.
The reciprocal logarithm function (Krämer, 2005)
has a deep connection with Euler's constant and was studied by James Gregory in connection with numerical integration. The coefficients C_{n} are called Gregory coefficients; the first six were given in a letter to John Collins in 1670. From the recursion
we get the table
n  1  2  3  4  5  6  7  8  9  10 

C_{n} 
Gregory coefficients are similar to Bernoulli numbers and satisfy the asymptotic relation
and the integral representation
Euler's constant has the integral representations
A very important expansion of Gregorio Fontana (1780) is:
which is convergent for all n.
Weighted sums of the Gregory coefficients give different constants:
The constant e^{γ} is important in number theory. Some authors denote this quantity simply as γ'. e^{γ} equals the following limit, where p_{n} is the nth prime number:
This restates the third of Mertens' theorems. The numerical value of e^{γ} is:
Other infinite products relating to e^{γ} include:
These products result from the Barnes Gfunction.
We also have
where the nth factor is the (n+1)st root of
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.
The continued fraction expansion of γ is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (sequence A002852 in OEIS), and has at least 470,000 terms.^{[3]}
Euler's generalized constants are given by
for 0 < α < 1, with γ as the special case α = 1.^{[4]} This can be further generalized to
for some arbitrary decreasing function f. For example,
gives rise to the Stieltjes constants, and
gives
where again the limit
appears.
A twodimensional limit generalization is the Masser–Gramain constant.
Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th22nd decimal places. (starting from the 20th digit, he calculated 1811209008239 when the correct value is 0651209008240.)
In recent decades, faster computers and algorithms have dramatically increased the number of calculated digits in the decimal expansion of γ.^{[5]}
Date  Decimal digits  Computation performed by 

1734  5  Leonhard Euler 
1736  15  Leonhard Euler 
1790  19  Lorenzo Mascheroni 
1809  22  Johann G. von Soldner 
1811  22  Carl Friedrich Gauss 
1812  40  Friedrich Bernhard Gottfried Nicolai 
1857  34  Christian Fredrik Lindman 
1861  41  Ludwig Oettinger 
1867  49  William Shanks 
1871  99  James W.L. Glaisher 
1871  101  William Shanks 
1878  263  John C. Adams 
1952  329  John William Wrench, Jr. 
1961  1050  Helmut Fischer and Carl Zeller 
1962  1,271  Donald Knuth 
1962  3,566  Dura W. Sweeney 
1973  4,879  William A. Beyer and Michael S. Waterman 
1977  20,700  Richard P. Brent 
1980  30,100  Richard P. Brent & Edwin M. McMillan 
1993  172,000  Jonathan Borwein 
1997  1,000,000  Thomas Papanikolaou 
December 1998  7,286,255  Xavier Gourdon 
October 1999  108,000,000  Xavier Gourdon & Patrick Demichel 
July 16, 2006  2,000,000,000  Shigeru Kondo & Steve Pagliarulo^{[6]} 
July 15, 2007  5,000,000,000  Shigeru Kondo & Steve Pagliarulo^{[7]} 
June 30, 2008  10,000,000,000  Shigeru Kondo & Steve Pagliarulo^{[8]} 
January 18, 2009  14,922,244,771  Alexander J. Yee & Raymond Chan^{[9]} 
March 13, 2009  29,844,489,545  Alexander J. Yee & Raymond Chan^{[10]} 
