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*

For more infomation of this nature, see Gourdon and Sebah (2004)

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
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