Euler–Mascheroni constant
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 gamma ().

It is defined as the limiting difference between the harmonic series and the natural logarithm:

Here,  represents the floor function.

The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:

History
The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations  and  for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations  and  for the constant. The notation  appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function .  For example, the German mathematician Carl Anton Bretschneider used the notation  in 1835  and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842

Appearances
The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):
 Expressions involving the exponential integral*
 The Laplace transform* of the natural logarithm
 The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*
 Calculations of the digamma function
 A product formula for the gamma function
 An inequality for Euler's totient function
 The growth rate of the divisor function
 In dimensional regularization of Feynman diagrams in quantum field theory
 The calculation of the Meissel–Mertens constant
 The third of Mertens' theorems*
 Solution of the second kind to Bessel's equation
 In the regularization/renormalization of the harmonic series as a finite value
 The mean of the Gumbel distribution
 The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
 The answer to the coupon collector's problem*
 In some formulations of Zipf's law
 A definition of the cosine integral*
 Lower bounds to a prime gap
 An upper bound on Shannon entropy in quantum information theory

Properties
The number  has not been proved algebraic or transcendental. In fact, it is not even known whether  is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if  is rational, its denominator must be greater than 10244663. The ubiquity of  revealed by the large number of equations below makes the irrationality of  a major open question in mathematics. Also see .

Relation to gamma function
 is related to the digamma function , and hence the derivative of the gamma function , when both functions are evaluated at 1. Thus:

This is equal to the limits:

Further limit results are :

A limit related to the beta function (expressed in terms of gamma functions) is

Relation to the zeta function
 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 . As a result, the formula is well-suited for efficient computation of the constant to high precision.

Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit :

and de la Vallée-Poussin's formula

where  are ceiling brackets.

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

where  is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, . Expanding some of the terms in the Hurwitz zeta function gives:

where 

 can also be expressed as follows where  is the Glaisher–Kinkelin constant:

 can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

Integrals
 equals the value of a number of definite integrals:

where  is the fractional harmonic number.

Definite integrals in which  appears include:

One can express  using a special case of Hadjicostas's formula as a double integral  and  with equivalent series:

An interesting comparison by  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 

where  and  are the number of 1s and 0s, respectively, in the base 2 expansion of .

We have also Catalan's 1875 integral (see )

Series expansions
In general, 

for any . However, the rate of convergence of this expansion depends significantly on . In particular,  exhibits much more rapid convergence than the conventional expansion  (; ). This is because

while

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the following infinite series approaches :

The series for  is equivalent to a series Nielsen found in 1897 (, ):

In 1910, Vacca found the closely related series (, , , , , , )

where  is the logarithm to base 2 and  is the floor function.

In 1926 he found a second series:

From the Malmsten–Kummer expansion for the logarithm of the gamma function  we get:

An important expansion for Euler's constant is due to Fontana and Mascheroni

where  are Gregory coefficients (, , ) This series is the special case  of the expansions

convergent for 

A similar series with the Cauchy numbers of the second kind  is (; ) 

Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series

where  are the Bernoulli polynomials of the second kind, which are defined by the generating function
 
For any rational  this series contains rational terms only. For example, at , it becomes
 
see  and . Other series with the same polynomials include these examples:
 
and
 
where  is the gamma function .

A series related to the Akiyama-Tanigawa algorithm is
 
where  are the Gregory coefficients of the second order .
  
Series of prime numbers:

Asymptotic expansions
 equals the following asymptotic formulas (where  is the th harmonic number):
 (Euler)
 (Negoi)
 (Cesàro)
The third formula is also called the Ramanujan expansion.

Exponential
The constant  is important in number theory. Some authors denote this quantity simply as .  equals the following limit, where  is the th prime number:

This restates the third of Mertens' theorems . The numerical value of  is:
 .

Other infinite products relating to  include:

These products result from the Barnes -function.

In addition,

where the th factor is the th root of

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow  using hypergeometric functions.

Continued fraction
The continued fraction expansion of  is of the form  , which has no apparent pattern. The continued fraction is known to have at least 475,006 terms, and it has infinitely many terms if and only if  is irrational.

Generalizations

Euler's generalized constants are given by

for , with  as the special case  . This can be further generalized to

for some arbitrary decreasing function . For example,

gives rise to the Stieltjes constants, and

gives

where again the limit

appears.

A two-dimensional limit generalization is the Masser–Gramain constant.

Euler–Lehmer constants are given by summation of inverses of numbers in a common
modulo class :

The basic properties are

and if  then

Published digits
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 20th–22nd and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.

Notes