## Euler–Mascheroni constant

 List of numbers – Irrational and suspected irrational numbers  $\gamma - \xi(3) - \sqrt{2} - \sqrt{3} - \sqrt{5} - \varphi - \rho - \delta_{S} - \alpha - e - \pi - \delta$ 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, … ] (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 $\gamma$ (gamma).

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

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

Its numerical value to 50 decimal places is

0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 … (sequence A001620 in OEIS).

$\gamma$ should not be confused with the base of the natural logarithm, $e$, which is sometimes called Euler’s number.

### History

The constant first appeared in a 1735 paper by the Swiss mathematician Leonard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascherori used the notations A and a for constant. The notation $\gamma$ 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 $\gamma$ in 1835.

### 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 Taylor series expansion for the Riemann zeta function *, where it is the first of the Stieltjes constants *
• Calculations of digamma function
• A product formula for the gamma function
• An inequality for Euler’s totient function
• The growth rate of the divisor function
• The calculation of the Meissel-Mertens constant
• The third of Mertens’ theorems*
• Solution of the second kind to Bessel’s equation
• In Dimansional regularization of Feynman diagrams in Quantum Field Theory
• The infomation entropy of the Weibull and Lévy distributions
• The answer to the Coupon collector’s problem*

### Properties

The number $\gamma$ has not been proved algebraic or transcendental. In fact, it is not even known whether $\gamma$ is irrational. Continued fraction analysis reveals that if $\gamma$ is rational, its denominator must be greater than $10^{242080}$. The ubiquity of $\gamma$ revealed by the large number of equations blow makes the irrationality of $\gamma$ a major open question in mathematics. Also see Sondow

For more quations of the sortshown below, see Gourdon and Sebeh (2002).

### Relation to gamma function

$\gamma$ is related to digamma function $\Psi$, and hence the derivative of the gamma function $\Gamma$, when both functions are evaluated at 1. Thus :
$-\gamma = \Gamma '\left(1\right) = \Psi\left(1\right)$

This is equal to limits :

$-\gamma = \lim_{z\rightarrow 0}{\left\{\Gamma\left(x\right)-\frac{1}{z}\right\}} = \lim_{z\rightarrow 0}{\left\{\Psi\left(z\right)+\frac{1}{z}\right\}}$.

Further limit results are (Krämer, 2005):

$\lim_{z\rightarrow 0}{\frac{1}{z}\left\{\frac{1}{\Gamma\left(1+z\right)}-\frac{1}{\Gamma\left(1-z\right)}\right\}} = 2\gamma\\ \lim_{z\rightarrow 0}{\frac{1}{z}\left\{\frac{1}{\Psi\left(1-z\right)}-\frac{1}{\Psi\left(1+z\right)}\right\}}$

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

$\gamma = \lim_{n\rightarrow \infty}{\left\{\frac{\Gamma\left(\frac{1}{n}\right)\Gamma\left(n+1\right)n^{1+\frac{1}{n}}}{\Gamma\left(2+n+\frac{1}{n}\right)}-\frac{n^2}{n+1}\right\}}.\\ \gamma = \lim_{m\rightarrow\infty}\sum_{k=1}^m{\binom{m}{k}\frac{\left(-1\right)^k}{k}ln\left(\Gamma\left(k+1\right)\right).}$

### Relation to zeta function

$\gamma$ can also be expressed as an infinite sum who se terms involve the Riemann zeta function evluated at positive intergers :

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

Other series related to the zeta function include :

$\gamma = \frac{3}{2}-ln2-\sum_{m=2}^\infty{\left(-1\right)^m\frac{m-1}{m}\left [ \zeta\left(m\right)-1\right ]}\\ = \lim_{n\rightarrow\infty}{\left [ \frac{2n-1}{2n}-ln n + \sum_{k=2}^n{\left( \frac{1}{k} - \frac{\zeta\left(1-k\right)}{n^k} \right)} \right ]}\\ = \lim_{n\rightarrow\infty}{\left [ \frac{2^n}{e^{2^n}}\sum_{m=0}^\infty{\frac{2^{mn}}{\left(m+1\right)!}}\sum_{t=0}^m{\frac{1}{t+1}} - n ln 2 + O\left(\frac{1}{2^ne^{2^n}}\right) \right ]}$

The error term in the last quation is a rapidly decreasing function of n. As a result, the formula is well-suited for effcient computation of the constant to hight precison.

Other interesting limits equaling the Euler-Mascheroni constant are the antisymmetric limit (Sondow, 1998)

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

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

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

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

Where $\xi\left(s,k\right)$ is the Hurwitz zeta function. The sum in this equation in volves the harmonic numbers, $H_n$. Expanding some of the terms in the Hurwits zeta function gives :

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

### Integrals

$\gamma$ equals the value of number of definite integrals :

$\gamma = -\int_0^\infty{e^{-x}lnxdx}\\ = -\int_0^1{lnln{\frac{1}{x}}dx}\\ = \int_0^\infty{\left(\frac{1}{e^x-1} - \frac{1}{xe^x}\right)dx} = \int_0^1{\left(\frac{1}{lnx} + \frac{1}{1-x}\right)dx}\\ = \int_0^\infty{\left(\frac{1}{1+x^k} - e^{-x}\right)\frac{dx}{x}},\,\,k > 0\\ = \int_0^1{H_xdx}$
where $H_x$ is the fractional Harmonic number.

Definite integrals in which $\gamma$ appears include :

$\int_0^\infty{e^{-x}lnxdx} = -\frac{1}{4}\left(\gamma+2ln2\right)\sqrt{\pi}\\ \int_0^\infty{e^{-x}{ln}^2xdx} = {\gamma}^2 + \frac{{\pi}^2}{6}. A081855$

One can express $\gamma$ using a special case of Hadjicostas’s formula as a double integral (Sondow 2003a, 2005) with equivalent series :

$\gamma = \int_0^1{\int_0^1{\frac{x-1}{\left(1-xy\right)ln\left(xy\right)}dxdy}} = \sum_{n=1}^\infty{\left(\frac{1}{n}-ln{\frac{n+1}{n}}\right).}$

An interesting comparison by J. Sondow (2005) is the double integral and alternating serres (A094640)
It shows that $ln{\left(\frac{4}{\pi}\right)}$ may be thought of as an “alternating Euler constant”.

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

$\sum_{n=1}^\infty{\frac{N_1\left(n\right) + N_1\left(n\right)}{2n\left(2n+1\right)}} = \gamma\\ \sum_{n=1}^\infty{\frac{N_1\left(n\right) + N_1\left(n\right)}{2n\left(2n+1\right)}} = ln{\left(\frac{4}{\pi}\right)}$
Where $N_1\left(n\right)$ and $N_0\left(n\right)$ 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)
$\gamma = \int_0^1{\frac{1}{1+x}\sum_{n=1}^\infty{x^{2^n-1}dx}}$.

### Series expansions

Euler showed that the fllowing infinite series appoaches $\gamma$ :
$\gamma = \sum_{k=1}^\infty{\left[ \frac{1}{k} - ln{\left(1 + frac{1}{k}\right)} \right].}$
The series for $\gamma$ is equivalent to series nielsen found in 1897 :
$\gamma = 1 - \sum_{k=2}^\infty{\left(-1\right)^k\frac{\left \lfloor \log_2k \right \rfloor}{k+1}}.$
In 1910, Vacca found the closely related series :
$\gamma = \sum_{k=2}^\infty\left( -1 \right )^k\frac{\left \lfloor \log_2k\right \rfloor}{k} = \frac{1}{2}-\frac{1}{3} + 2\left ( \frac{1}{4} - \frac{1}{5}+\frac{1}{6} - \frac{1}{7} \right )+3\left (\frac{1}{8}-\frac{1}{9} + \frac{1}{10}-\frac{1}{11} +\dots - \frac{1}{15} \right )+\dots$
where $\log_2$ is the logarithm to base 2 and $\left \lfloor \right \rfloor$ is the floor function.
In 1926 he found a second series :
$\gamma + \zeta\left(2 \right ) = \sum_{k=2}^\infty{\left(\frac{1}{{\left \lfloor \sqrt{k} \right \rfloor}^2} - \frac{1}{k} \right )} = \sum_{k=2}^\infty{\frac{k-{\left \lfloor \sqrt{k} \right \rfloor}^2}{k{\left \lfloor \sqrt{k} \right \rfloor}^2}} = \frac{1}{2}+\frac{2}{3}+\frac{1}{2^2}\sum_{k=1}^{2\times2}{\frac{k}{k+2^2}}+\frac{1}{3^2}\sum_{k=1}^{3\times2}{\frac{k}{k+3^2}}+\dots$
From the Kummer-expansion of the gamma function we get :
$\gamma = \ln{\pi} - 4\ln{\Gamma\left (\frac{3}{4} \right )} + \frac{4}{\pi}\sum_{k=1}^\infty{\left(-1 \right )^{k+1}\frac{\ln{\left ( 2k+1 \right )}}{2k+1}}$
Series of prime numbers :
$\gamma = \lim_{n\rightarrow\infty}{\left (\ln{n}-\sum_{p\leq n}{\frac{\ln{p}}{p-1}} \right )}$

### Asymptotic expansions

$\gamma$ equals the fllowing asymptotic formulas (where $H_n$ is the nth harmonic number.)
$\gamma \sim H_n-\ln{\left (n \right )} - \frac{1}{2n}+\frac{1}{12n^2}-\frac{1}{120n^4}+\dots (Euler)$
$\gamma \sim H_n-ln{\left (n + \frac{1}{2}+\frac{1}{24n}-\frac{1}{48n^3}+\dots \right )} (Negoi)$
$\gamma \sim H_n-\frac{\ln{\left(n \right )}+\ln{\left(n+1 \right )}}{2} - \frac{1}{6n\left(n+1 \right )} + \frac{1}{30n^2\left(n+1 \right )^2} -\dots (Cesaro)$
The third formula is also called the Ramanujan expansion.

### Relations with the reciprocal logarithm

The reciprocal logarithm function (Krämer, 2005)
$\frac{z}{\ln{\left(1-z \right )}}=\sum_{n=0}^\infty{C_nz^n}, \,\,\left | z \right | < 1,$
has a deep connection with Euler’s constant and was studied by James Gregory in connection with numerical intergration. The coefficients $C_n$ are called Gregory coefficients; the first six were given in a latter to John Collins in 1670. From the recursin
$C_0 = -1, \,\, \sum_{k=0}^{n-1}{\frac{C_k}{n-k}}=0, \,\,n=2,3,4,\dots,$
we get the table (A002206 and A002207)

 n 1 2 3 4 5 6 7 8 9 10 $C_n$ $\frac{1}{2}$ $\frac{1}{12}$ $\frac{1}{24}$ $\frac{19}{720}$ $\frac{3}{160}$ $\frac{863}{60480}$ $\frac{275}{24192}$ $\frac{33953}{3628800}$ $\frac{8183}{1036800}$ $\frac{3250433}{479001600}$

Gregory coefficients are similar to Bernoulli numbers and satisfy the asymptotic relation
$C_n = \frac{1}{n\ln^2{n}}-O\left(\frac{1}{n\ln^3{n}} \right ),\,\,n\rightarrow\infty$
and the integral representation
$C_n = \int_0^\infty{\frac{dx}{\left(1+x \right )^n\left(\ln^2{x}+\pi^2 \right )}}, n=1,2,\dots$
Euler’s constant has the integral representations
$\int_0^\infty {\frac{\ln{\left (1+x \right )}}{\ln^2{x}+\pi^2}.\frac{dx}{x^2}} = \int_{-\infty}^{\infty}{\frac{\ln{\left (1+e^{-x} \right )}}{x^2+\pi^2}e^xdx}$
A very important expansion of Gregorio Fontana (1780) is:
$H_n = \gamma+\log{n}+\frac{1}{2n}-\sum_{k=2}^{\infty}{\frac{\left ( k-1 \right)!C_k}{n\left (n+1 \right )\dots\left (n+k-1 \right )}},\,\,n=1,2,\dots,\\ = \gamma + \log{n}+\frac{1}{2n}-\frac{1}{12n\left (n+1 \right )}-\frac{1}{12n\left (n+1 \right )\left (n+2 \right )}-\frac{19}{120n\left (n+1 \right )\left ( n+2 \right )\left ( n+3 \right )}-\dots$
which is convergent for all n.
Weighted sums of the Gregory coefficients give different constants:
$1 = \sum_{n=1}^\infty{C_n} = \frac{1}{2}+\frac{1}{12}+\frac{1}{24}+\frac{19}{720}+\frac{3}{160}+\dots,\\ \frac{1}{\log{2}}-1 = \sum_{n=1}^\infty{\left (-1 \right )^{n+1}C_n} = \frac{1}{2}-\frac{1}{12}+\frac{1}{24}-\frac{19}{720}+\frac{3}{160}-\dots,\\ \gamma = \sum_{n=1}^\infty{\frac{C_n}{n}}= \frac{1}{2}+\frac{1}{24}+\frac{1}{72}+\frac{19}{2880}+\frac{3}{800}+\dots$

### e$\gamma$

The constant $e^\gamma$ is important in number theory. Some authors denote this quantity simply as $\gamma '. e^\gamma$ equals the following limit, where $p_n$ is the n-th prime number :
$e^\gamma = \lim_{n\rightarrow \infty}{\frac{1}{\ln{p_n}}\prod_{i=1}^{n}{\frac{p_i}{p_i-1}}.}$
This restates the third of Mertens’ theorems. The numerical value of $e^\gamma$ is A073004
$e^\gamma = 1.78107241799019798523650410310717954916964521430343\dots$
Other infinite products relating to $e^\gamma$ include :
$\frac{e^{1+\frac{\gamma}{2}}}{\sqrt{2\pi}} = \prod_{n=1}^{\infty}{e^{-1+\frac{1}{2n}}\left (1 + \frac{1}{n} \right )^n}\\ \frac{e^{3+2\gamma}}{\sqrt{2\pi}} = \prod_{n=1}^{\infty}{e^{-2+\frac{2}{n}}\left (1 + \frac{2}{n} \right )^n}$
These products result from the Barnes G-function.
We also have
$e^\gamma = \left ( \frac{2}{1}\right)^{\frac{1}{2}}\left (\frac{2^2}{1.3} \right )^{\frac{1}{3}}\left (\frac{2^3.4}{1.3^3} \right )^{\frac{1}{4}}\left (\frac{2^4.4^4}{1.3^6.5} \right )^{\frac{1}{5}}\dots$
where the n-th factor is the (n+1)-st root of
$\prod_{k=1}^n{\left (k+1 \right )^{\left (-1 \right )^{k+1}\binom{n}{k}}}$
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.

### Continued fraction

The continued fraction expansion of $\gamma$ is of the form $\left [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,\dots \right ]$ (sequence A002852 in OEIS), and has at least 470,000 terms.

### Generalizations

Euler’s generalized constants are given by $\gamma_\alpha = \lim_{n\rightarrow\infty}{\left [\sum_{k=1}^n{\frac{1}{k^\alpha}-\int_1^n{\frac{1}{x^\alpha}dx}} \right ],}$ for $0 < \alpha < 1,$ with $\gamma$ as the special case $\alpha = 1.$ This can be further generalized to
$c_f = \lim_{n\rightarrow\infty}{\left [\sum_{k=1}^n{f\left (k \right )-\int_1^n{f(x)dx}} \right ]}$
for some arbitrary decreasing function f. For example,
$f_n(x)=\frac{\ln^nx}{x}$
gives rise to the Stieltjes constants, and
$f_a(x) = x^{-a}$ gives $\gamma_{f_a} = \frac{(a-1)\zeta(a)-1}{a-1}$ appears.
A two-dimensional limit generalization is the Masser-Gramain constant.

### 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 decimal places. (starting from the 20th digit, he calculated 1811209008239 when the correct value is 0651209008240.)

#### Published Decimal Expansions of $\gamma$

 Date Decimal digits Author 1734 5 Leonahard Euler 1736 15 Leonahard Euler 1790 19 Lorenzo Mascheroni 1811 22 Carl Friedrich Gauss 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 2009 29,844,489,545 Alexander J. Yee & Raymond Chan