Euler's constant

"Euler's constant" redirects here. For the base of the natural logarithm, e ≈ 2.718..., see e (mathematical constant).

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 γ (gamma).

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

$\gamma =\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)\right)=\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx.$

Its numerical value to 50 decimal places is

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

$\gamma$ should not to 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 Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations 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.

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 the 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 Dimensional regularization of Feynman diagrams in Quantum Field Theory.
• The mean of the Gumbel distribution.

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

Properties

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 10²⁴²⁰⁸⁰. 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).

Relation to gamma function

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:

$\ -\gamma =\Gamma '(1)=\Psi (1).$

This is equal to the limits:

$-\gamma =\lim _{z\to 0}\left\{\Gamma (z)-{\frac {1}{z}}\right\}=\lim _{z\to 0}\left\{\Psi (z)+{\frac {1}{z}}\right\}.$

Further limit results are (Krämer, 2005):

$\lim _{z\to 0}{\frac {1}{z}}\left\{{\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right\}=2\gamma$

$\lim _{z\to 0}{\frac {1}{z}}\left\{{\frac {1}{\Psi (1-z)}}-{\frac {1}{\Psi (1+z)}}\right\}={\frac {\pi ^{2}}{3\gamma ^{2}}}.$

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

$\gamma =\lim _{n\to \infty }\left\{{\frac {\Gamma ({\frac {1}{n}})\Gamma (n+1)\,n^{1+1/n}}{\Gamma (2+n+{\frac {1}{n}})}}-{\frac {n^{2}}{n+1}}\right\}.$

$\gamma =\lim \limits _{m\to \infty }\sum _{k=1}^{m}{m \choose k}{\frac {(-1)^{k}}{k}}\ln(\Gamma (k+1)).$

Relation to the zeta function

γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

${\begin{aligned}\gamma &=\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\ln \left({\frac {4}{\pi }}\right)+\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{2^{m-1}m}}.\end{aligned}}$

Other series related to the zeta function include:

${\begin{aligned}\gamma &={\frac {3}{2}}-\ln 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}[\zeta (m)-1]\\&=\lim _{n\to \infty }\left[{\frac {2\,n-1}{2\,n}}-\ln \,n+\sum _{k=2}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right]\\&=\lim _{n\to \infty }\left[{\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{m\,n}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\,\ln 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right].\end{aligned}}$

The error term in the last equation is a rapidly decreasing function of n. 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 (Sondow, 1998)

$\gamma =\lim _{s\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{s}}}-{\frac {1}{s^{n}}}\right)=\lim _{s\to 1}\left(\zeta (s)-{\frac {1}{s-1}}\right)$

and

${\begin{aligned}\gamma =\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right).\end{aligned}}$

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:

$\gamma =\sum _{k=1}^{n}{\frac {1}{k}}-\ln n-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}$

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:

$H_{n}=\ln n+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\varepsilon$, where $0<\varepsilon <{\frac {1}{252n^{6}}}.$

Integrals

γ equals the value of a number of definite integrals:

${\begin{aligned}\gamma &=-\int _{0}^{\infty }{e^{-x}\ln x}\,dx\\&=-\int _{0}^{1}\ln \ln \left({\frac {1}{x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)dx=\int _{0}^{1}\left({\frac {1}{\ln x}}+{\frac {1}{1-x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{1+x^{k}}}-e^{-x}\right){\frac {dx}{x}},\quad k>0.\end{aligned}}$

Definite integrals in which γ appears include:

$\int _{0}^{\infty }{e^{-x^{2}}\ln x}\,dx=-{\tfrac {1}{4}}(\gamma +2\ln 2){\sqrt {\pi }}$

$\int _{0}^{\infty }{e^{-x}\ln ^{2}x}\,dx=\gamma ^{2}+{\frac {\pi ^{2}}{6}}.$

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

$\gamma =\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-x\,y)\ln(x\,y)}}\,dx\,dy=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right).$

An interesting comparison by J. Sondow (2005) is the double integral and alternating series

$\ln \left({\frac {4}{\pi }}\right)=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+x\,y)\ln(x\,y)}}\,dx\,dy=\sum _{n=1}^{\infty }(-1)^{n-1}\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right).$

It shows that $\ln \left({\frac {4}{\pi }}\right)$ may be thought of as an "alternating Euler constant".

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

$\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}=\gamma$

$\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}}=\ln \left({\frac {4}{\pi }}\right)$

where N₁(n) and N₀(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)

$\gamma =\int _{0}^{1}{\frac {1}{1+x}}\sum _{n=1}^{\infty }x^{2^{n}-1}\,dx.$

Series expansions

Euler showed that the following infinite series approaches γ:

$\gamma =\sum _{k=1}^{\infty }\left[{\frac {1}{k}}-\ln \left(1+{\frac {1}{k}}\right)\right].$

The series for $\gamma$ is equivalent to series Nielsen found in 1897:

$\gamma =1-\sum _{k=2}^{\infty }(-1)^{k}{\frac {\lfloor \log _{2}k\rfloor }{k+1}}$.

In 1910, Vacca found the closely related series:

${\gamma =\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\frac {1}{2}}-{\frac {1}{3}}+2\left({\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{6}}-{\frac {1}{7}}\right)+3\left({\frac {1}{8}}-{\frac {1}{9}}+{\frac {1}{10}}-{\frac {1}{11}}+\dots -{\frac {1}{15}}\right)+\dots }$

where $\log _{2}$ is the logarithm to the base 2 and $\lfloor \,\rfloor$ is the floor function.

In 1926 he found a second series:

${\gamma +\zeta (2)=\sum _{k=2}^{\infty }\left({\frac {1}{\lfloor {\sqrt {k}}\rfloor ^{2}}}-{\frac {1}{k}}\right)=\sum _{k=2}^{\infty }{\frac {k-\lfloor {\sqrt {k}}\rfloor ^{2}}{k\lfloor {\sqrt {k}}\rfloor ^{2}}}={\frac {1}{2}}+{\frac {2}{3}}+{\frac {1}{2^{2}}}\sum _{k=1}^{2\times 2}{\frac {k}{k+2^{2}}}+{\frac {1}{3^{2}}}\sum _{k=1}^{3\times 2}{\frac {k}{k+3^{2}}}+\dots }$

From the Kummer-expansion of the gamma function we get:

$\gamma =\ln \pi -4\ln \Gamma ({\tfrac {3}{4}})+{\frac {4}{\pi }}\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\ln(2k+1)}{2k+1}}$

Asymptotic expansions

γ equals the following asymptotic formulas (where $H_{n}$ is the nth harmonic number.)

$\gamma \sim H_{n}-\ln \left(n\right)-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+...$

(Euler)

$\gamma \sim H_{n}-\ln \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{3}}}+...}\right)$

(Negoi)

$\gamma \sim H_{n}-{\frac {\ln \left(n\right)+\ln \left({n+1}\right)}{2}}-{\frac {1}{6n\left({n+1}\right)}}+{\frac {1}{30n^{2}\left({n+1}\right)^{2}}}-...$

(Cesaro)

The third formula is also called the Ramanujan expansion.

Relations with the reciprocal logarithm

The reciprocal logarithm function (Krämer, 2005)

${\frac {z}{\ln(1-z)}}=\sum _{n=0}^{\infty }C_{n}z^{n},\quad |z|<1,$

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

$C_{0}=-1,\quad \sum _{k=0}^{n-1}{\frac {C_{k}}{n-k}}=0,\quad n=2,3,4,\dots ,$

we get the table

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

Gregory coefficients are similar to Bernoulli numbers and satisfy the asymptotic relation

$C_{n}={\frac {1}{n\ln ^{2}n}}-{\mathcal {O}}\left({\frac {1}{n\ln ^{3}n}}\right),\quad n\to \infty ,$

and the integral representation

$C_{n}=\int _{0}^{\infty }{\frac {dx}{(1+x)^{n}\left(\ln ^{2}x+\pi ^{2}\right)}},\quad n=1,2,\dots .$

Euler's constant has the integral representations

$\gamma =\int _{0}^{\infty }{\frac {\ln(1+x)}{\ln ^{2}x+\pi ^{2}}}\cdot {\frac {dx}{x^{2}}}=\int _{-\infty }^{\infty }{\frac {\ln(1+e^{-x})}{x^{2}+\pi ^{2}}}\,e^{x}\,dx.$

A very important expansion of Gregorio Fontana (1780) is:

${\begin{aligned}H_{n}&=\gamma +\log n+{\frac {1}{2n}}-\sum _{k=2}^{\infty }{\frac {(k-1)!C_{k}}{n(n+1)\dots (n+k-1)}},\quad n=1,2,\dots ,\\&=\gamma +\log n+{\frac {1}{2n}}-{\frac {1}{12n(n+1)}}-{\frac {1}{12n(n+1)(n+2)}}-{\frac {19}{120n(n+1)(n+2)(n+3)}}-\dots \end{aligned}}$

which is convergent for all n.

Weighted sums of the Gregory coefficients give different constants:

${\begin{aligned}1&=\sum _{n=1}^{\infty }C_{n}={\tfrac {1}{2}}+{\tfrac {1}{12}}+{\tfrac {1}{24}}+{\tfrac {19}{720}}+{\tfrac {3}{160}}+\dots ,\\{\frac {1}{\log 2}}-1&=\sum _{n=1}^{\infty }(-1)^{n+1}C_{n}={\tfrac {1}{2}}-{\tfrac {1}{12}}+{\tfrac {1}{24}}-{\tfrac {19}{720}}+{\tfrac {3}{160}}-\dots ,\\\gamma &=\sum _{n=1}^{\infty }{\frac {C_{n}}{n}}={\tfrac {1}{2}}+{\tfrac {1}{24}}+{\tfrac {1}{72}}+{\tfrac {19}{2880}}+{\tfrac {3}{800}}+\dots .\end{aligned}}$

e^γ

The constant e^γ is important in number theory. Some authors denote this quantity simply as $\gamma ^{\prime }$. e^γ equals the following limit, where p_n is the n-th prime number:

$e^{\gamma }=\lim _{n\to \infty }{\frac {1}{\ln p_{n}}}\prod _{i=1}^{n}{\frac {p_{i}}{p_{i}-1}}.$

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

$e^{\gamma }=1.78107241799019798523650410310717954916964521430343\dots$

Other infinite products relating to e^γ include:

${\frac {e^{1+\gamma /2}}{\sqrt {2\,\pi }}}=\prod _{n=1}^{\infty }e^{-1+1/(2\,n)}\,\left(1+{\frac {1}{n}}\right)^{n}$

${\frac {e^{3+2\gamma }}{2\,\pi }}=\prod _{n=1}^{\infty }e^{-2+2/n}\,\left(1+{\frac {2}{n}}\right)^{n}.$

These products result from the Barnes G-function.

We also have

$e^{\gamma }=\left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/3}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/4}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/5}\cdots$

where the nth factor is the (n+1)st root of

$\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}.$

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 [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (sequence A002852 in the OEIS), and has at least 470,000 terms.

Generalizations

Euler's generalized constants are given by

$\gamma _{\alpha }=\lim _{n\to \infty }\left[\sum _{k=1}^{n}{\frac {1}{k^{\alpha }}}-\int _{1}^{n}{\frac {1}{x^{\alpha }}}\,dx\right],$

for 0 < α < 1, with γ as the special case α = 1. This can be further generalized to

$c_{f}=\lim _{n\to \infty }\left[\sum _{k=1}^{n}f(k)-\int _{1}^{n}f(x)\,dx\right]$

for some arbitrary decreasing function f. For example,

$f_{n}(x)={\frac {\ln ^{n}x}{x}}$

gives rise to the Stieltjes constants, and

$f_{a}(x)=x^{-a}$

gives

$\gamma _{f_{a}}={\frac {(a-1)\zeta (a)-1}{a-1}}$

where again the limit

$\gamma =\lim _{a\to 1}\left[\zeta (a)-{\frac {1}{a-1}}\right]$

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 γ

Date	Decimal digits	Author
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
2009	29,844,489,545	Alexander J. Yee & Raymond Chan

References

• Borwein, Jonathan M., David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". Journal of Computational and Applied Mathematics. 121: 11. {{cite journal}}: External link in |title= (help)CS1 maint: multiple names: authors list (link) Derives γ as sums over Riemann zeta functions.
• Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ."
• ----- (2004) "The Euler constant: γ."
• Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. ISBN 0-201-89683-4
• Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Universität Göttingen.
• Sondow, Jonathan (1998) "An antisymmetric formula for Euler's constant," Mathematics Magazine 71: 219-220.
• ------ (2002) "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant." With an Appendix by Sergey Zlobin, Mathematica Slovaca 59: 307-314.
• ------ (2003) "An infinite product for e^γ via hypergeometric formulas for Euler's constant, γ."
• ------ (2003a) "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131: 3335-3344.
• ------ (2005) "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula," American Mathematical Monthly 112: 61-65.
• ------ (2005) "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π."
• ------ and Wadim Zudilin (2006), "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper," Ramanujan Journal 12: 225-244.
• G. Vacca (1926), "Nuova serie per la costante di Eulero, C = 0,577…". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche, Matematiche e Naturali (6) 3, 19–20.
• James Whitbread Lee Glaisher (1872), "On the history of Euler's constant". Messenger of Mathematics. New Series, vol.1, p.25-30, JFM 03.0130.01
• Carl Anton Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle Journal, vol.17, p.257-285 (submitted 1835)
• Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.
• Lorenzo Mascheroni (1792). "Adnotationes ad calculum integralem Euleri. In quibus nonnullae formulae ab Eulero propositae evolvuntur". Galeati, Ticini. Both online at: http://books.google.de/books?id=XkgDAAAAQAAJ
• Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 0-691-09983-9.

External links

• Weisstein, Eric W. "Euler-Mascheroni constant". MathWorld.
• Krämer, Stefan "Euler's Constant γ=0.577... Its Mathematics and History."
• Jonathan Sondow.