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 γ (gamma).

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

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

Here, ⌊x⌋ represents the floor function. The numerical value of this constant, to 50 decimal places, is

0.57721566490153286060651209008240243104215933593992… (sequence A001620 in OEIS).

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

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 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] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[3]
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
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*
A definition of the cosine integral*

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.[4] 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:

\( -\gamma = \lim_{z\to 0} \left\{\Gamma(z) - \frac1{z} \right\} = \lim_{z\to 0} \left\{\Psi(z) + \frac1{z} \right\}. \)

Further limit results are (Krämer, 2005):

\( \lim_{z\to 0} \frac1{z}\left\{\frac1{\Gamma(1+z)} - \frac1{\Gamma(1-z)} \right\} = 2\gamma \)
\( \lim_{z\to 0} \frac1{z}\left\{\frac1{\Psi(1-z)} - \frac1{\Psi(1+z)} \right\} = \frac{\pi^2}{3\gamma^2}. \)

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:

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

Other series related to the zeta function include:

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

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

\( \begin{align} \gamma = \lim_{n \to \infty} \frac{1}{n}\, \sum_{k=1}^n \left ( \left \lceil \frac{n}{k} \right \rceil - \frac{n}{k} \right ).\end{align} \)

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, Hn. 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:

\( \begin{align}\gamma &= - \int_0^\infty { e^{-x} \ln x }\,dx \\ &= -\int_0^1 \ln\ln\left (\frac{1}{x}\right) dx \\ &= \int_0^\infty \left (\frac1{e^x-1}-\frac1{xe^x} \right)dx = \int_0^1\left(\frac 1{\ln x} + \frac 1{1-x}\right)dx\\ &= \int_0^\infty \left (\frac1{1+x^k}-e^{-x} \right)\frac{dx}{x},\quad k>0\\ &= \int_0^1 H_{x} dx \end{align} \)

where \( H_{x} is the fractional Harmonic number.

Definite integrals in which γ appears include:

\( \int_0^\infty { e^{-x^2} \ln x }\,dx = -\tfrac14(\gamma+2 \ln 2) \sqrt{\pi} \)

\( \int_0^\infty { e^{-x} \ln^2 x }\,dx = \gamma^2 + \frac{\pi^2}{6} . \)

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 \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}(-1)^k\frac{\lfloor\log_2 k\rfloor}{k+1}. \)

In 1910, Vacca found the closely related series:

\( { \gamma = \sum_{k=2}^\infty (-1)^k \frac{ \left \lfloor \log_2 k \right \rfloor}{k} = \frac12-\frac13 + 2\left(\frac14 - \frac15 + \frac16 - \frac17\right) + 3\left(\frac18 - \frac19 + \frac1{10} - \frac1{11} + \dots - \frac1{15}\right) + \dots } \)

where \( \log_2 \) is the logarithm to the base 2 and \( \lfloor \, \rfloor \) is the floor function.

In 1926 he found a second series:

\( {\gamma + \zeta(2) = \sum_{k=2}^\infty\left(\frac1{\lfloor \sqrt{k} \rfloor^2} - \frac1{k}\right) = \sum_{k=2}^{\infty} \frac{k - \lfloor\sqrt{k}\rfloor^2}{k\lfloor\sqrt{k}\rfloor^2} = \frac12 + \frac23 + \frac1{2^2} \sum_{k=1}^{2 \times 2} \frac k {k+2^2} + \frac1{3^2} \sum_{k=1}^{3 \times 2} \frac k {k+3^2} + \dots}. \)

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

\( \gamma = \ln\pi - 4\ln\Gamma(\tfrac34) + \frac4{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\ln(2k+1)}{2k+1}. \)

Series of prime numbers:

\( \begin{align} \gamma = \lim_{n \to \infty} \left( \ln n - \sum_{p \le n} \frac{ \ln p }{ p-1 } \right)\end{align}. \) [5]

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)

\( \frac{z}{\ln(1-z)} = \sum_{n=0}^{\infty}C_nz^n, \quad |z|<1, \)

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	OEIS sequence
Cn	\( \tfrac12 \)	\( \tfrac1{12}\)	\( \tfrac1{24} \)	\( \tfrac{19}{720}\)	\( \tfrac3{160}\)	\( tfrac{863}{60480}\)	\( \tfrac{275}{24192}\)	\( \tfrac{33953}{3628800}\)	\(\tfrac{8183}{1036800} \)	\( \tfrac{3250433}{479001600}\)	A002206 (numerators), A002207 (denominators)

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

\( C_n = \frac1{n\ln^2 n} - \mathcal{O}\left(\frac1{n\ln^3 n}\right),\quad n\to\infty, \)

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:

\( \begin{align} H_n &= \gamma + \log n + \frac1{2n} - \sum_{k=2}^{\infty}\frac{(k-1)!C_k}{n(n+1)\dots(n+k-1)},\quad n=1,2,\dots,\\ &= \gamma + \log n + \frac1{2n} - \frac1{12n(n+1)} - \frac1{12n(n+1)(n+2)} - \frac{19}{120n(n+1)(n+2)(n+3)} - \dots \end{align} \)

which is convergent for all n.

Weighted sums of the Gregory coefficients give different constants:

\( \begin{align} 1 &= \sum_{n=1}^{\infty}C_n = \tfrac12 + \tfrac1{12} + \tfrac1{24} + \tfrac{19}{720} + \tfrac3{160} + \dots,\\ \frac1{\log2} - 1 &= \sum_{n=1}^{\infty}(-1)^{n+1}C_n = \tfrac12 - \tfrac1{12} + \tfrac1{24} - \tfrac{19}{720} + \tfrac3{160} - \dots,\\ \gamma &= \sum_{n=1}^{\infty}\frac{C_n}{n} = \tfrac12 + \tfrac1{24} + \tfrac1{72} + \tfrac{19}{2880} + \tfrac3{800} + \dots. \end{align} \)

eγ

The constant eγ is important in number theory. Some authors denote this quantity simply as \( \gamma^\prime \) . eγ equals the following limit, where pn is the nth 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 eγ is:

1.78107241799019798523650410310717954916964521430343 … (sequence A073004 in OEIS).

Other infinite products relating to eγ 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 OEIS), and has at least 470,000 terms.[4]
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.[6] 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

\( \gamma = \lim_{a\to1}\left[ \zeta(a) - \frac{1}{a-1}\right] \)

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
1735	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
1877	262	J. C. Adams
1952	328	John William Wrench, Jr.
1961	1050	Helmut Fischer and Karl 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[7]

Notes

^ Sloane's A002852 : Continued fraction for Euler's constant. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Carl Anton Bretschneider: Theoriae logarithmi integralis lineamenta nova (13 October 1835), Journal für die reine und angewandte Mathematik 17, 1837, pp. 257–285 (in Latin; “γ = c = 0,577215 664901 532860 618112 090082 3..” on p. 260)
^ Augustus De Morgan: The differential and integral calculus, Baldwin and Craddock, London 1836–1842 (“γ” on p. 578)
^ a b Havil 2003 p 97.
^ http://mathworld.wolfram.com/MertensConstant.html (15)
^ Havil, 117-118
^ Nagisa – Large Computations

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. 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 eγ 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.
E.A. Karatsuba, Fast evaluation of transcendental functions. Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991
E.A. Karatsuba, On the computation of the Euler constant γ, J. of Numerical Algorithms Vol.24, No.1-2, pp. 83–97 (2000)

External links

Weisstein, Eric W., "Euler-Mascheroni constant" from MathWorld.
Krämer, Stefan "Euler's Constant γ=0.577... Its Mathematics and History."
Jonathan Sondow.
Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)

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