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| {{redirect|Euler's constant|the base of the natural logarithm, e ≈ 2.718...|e (mathematical constant)}}
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| [[File:gamma-area.svg|thumb|The area of the blue region converges on the Euler–Mascheroni constant.]]
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| The '''Euler–Mascheroni constant''' (also called '''Euler's constant''') is a [[mathematical constant]] recurring in [[mathematical analysis|analysis]] and [[number theory]], usually denoted by the lowercase Greek letter {{lang|el|gamma}} (<math>\gamma</math>).
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| It is defined as the [[limit of a sequence|limiting]] difference between the [[harmonic series (mathematics)|harmonic series]] and the [[natural logarithm]]:
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| :<math>\gamma = \lim_{n \rightarrow \infty } \left(
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| \sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx.</math>
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| Here, <math>\lfloor x\rfloor</math> represents the [[floor and ceiling functions|floor function]].
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| The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is
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| : {{gaps|0.57721|56649|01532|86060|65120|90082|40243|10421|59335|93992|…}} {{OEIS2C|id=A001620}}.
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| {| class="infobox" style ="width: 370px;"
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| | colspan="2" align="center" |{{Irrational numbers}}
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| |-
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| |[[Binary numeral system|Binary]]
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| |{{gaps|0.1001001111000100011001111110001101111101...}}
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| |-
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| |[[Decimal]]
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| |{{gaps|0.5772156649015328606065120900824024310421...}}
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| |-
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| |[[Hexadecimal]]
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| |{{gaps|0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3A...}}
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| |-
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| |[[Continued fraction]]
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| |{{nowrap|[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ]}}<ref name="ReferenceA">{{OEIS2C|id=A002852|name=Continued fraction for Euler's constant}}</ref><small>(This continued fraction is not known to be finite or [[periodic function|periodic]]. Shown in [[continued fraction#Notations for continued fractions|linear notation]])</small>
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| |}
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| ==History==
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| The constant first appeared in a 1734 paper by the [[Ancien Régime of Switzerland|Swiss]] mathematician [[Leonhard Euler]], titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43). Euler used the notations ''C'' and ''O'' for the constant. In 1790, [[Italy|Italian]] mathematician [[Lorenzo Mascheroni]] used the notations ''A'' and ''a'' for the constant. The notation <math>\gamma</math> 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]].<ref name=lagarias>{{Cite journal| last = Lagarias | first = Jeffrey C. |date=Octoberl 2013 | title = Euler's constant: Euler's work and modern developments | journal = [[Bulletin of the American Mathematical Society]] | volume = 50 | issue = 4 | page = 556 | url = http://www.ams.org/journals/bull/2013-50-04/S0273-0979-2013-01423-X/S0273-0979-2013-01423-X.pdf | format = PDF }}</ref> For example, the German mathematician [[Carl Anton Bretschneider]] used the notation <math>\gamma</math> in 1835<ref>[[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 [http://books.google.de/books?id=OAoPAAAAIAAJ&pg=PA260 p. 260])</ref> and [[Augustus De Morgan]] used it in a textbook published in parts from 1836 to 1842.<ref>[[Augustus De Morgan]]: ''The differential and integral calculus'', Baldwin and Craddock, London 1836–1842 (“''γ''” on [http://books.google.com/books?id=95x4IrIcHrgC&pg=PA578 p. 578])</ref> | |
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| ==Appearances==
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| The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):
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| * Expressions involving the [[exponential integral]]*
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| * The [[Laplace transform]]* of the [[natural logarithm]]
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| * The first term of the [[Taylor series]] expansion for the [[Riemann zeta function]]*, where it is the first of the [[Stieltjes constants]]*
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| * Calculations of the [[digamma function]]
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| * A product formula for the [[gamma function]]
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| * An inequality for [[Euler's totient function]]
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| * The growth rate of the [[divisor function]]
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| * The calculation of the [[Meissel–Mertens constant]]
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| * The third of [[Mertens' theorems]]*
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| * Solution of the second kind to [[Bessel's equation]]
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| * In the regularization/[[renormalization]] of the Harmonic series as a finite value
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| * In [[Dimensional regularization]] of [[Feynman diagram]]s in [[Quantum Field Theory]]
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| * The [[mean]] of the [[Gumbel distribution]]
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| * The [[information entropy]] of the [[Weibull distribution|Weibull]] and [[Lévy distribution|Lévy]] distributions, and, implicitly, of the [[chi-squared distribution]] for one or two degrees of freedom.
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| * The answer to the [[coupon collector's problem]]*
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| * In some formulations of [[Zipf's law]]
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| * A definition of the [[trigonometric integral#Cosine integral|cosine integral]]*
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| * In expressions revealing the key properties of an [[exoplanet]] atmosphere (temperature, pressure, and composition) embedded in its [[absorption spectrum]], which are at the basis of a new method to determine the mass of exoplanets, ''MassSpec''.<ref>{{cite journal|last=de Wit|first=Julien|coauthors=Seager, S.|title=Constraining Exoplanet Mass from Transmission Spectroscopy|journal=Science|date=19 December 2013|volume=342|issue=6165|pages=1473–1477|doi=10.1126/science.1245450|url=http://www.sciencemag.org/content/342/6165/1473}}</ref>
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| For more information of this nature, see [http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Gourdon and Sebah (2004).]
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| ==Properties==
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| The number <math>\gamma</math> has not been proved [[algebraic number|algebraic]] or [[transcendental number|transcendental]]. In fact, it is not even known whether <math>\gamma</math> is [[irrational number|irrational]]. [[Continued fraction]] analysis reveals that if <math>\gamma</math> is [[rational number|rational]], its denominator must be greater than 10<sup>242080</sup>.<ref name=Havil /> The ubiquity of <math>\gamma</math> revealed by the large number of equations below makes the irrationality of <math>\gamma</math> a major open question in mathematics. Also see Sondow (2003a).
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| For more equations of the sort shown below, see [http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Gourdon and Sebah (2002).]
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| ===Relation to gamma function===
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| <math>\gamma</math> is related to the [[digamma function]] Ψ, and hence the derivative of the [[gamma function]] Γ, when both functions are evaluated at 1. Thus:
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| :<math> \ -\gamma = \Gamma'(1) = \Psi(1). </math>
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| This is equal to the limits:
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| :<math> -\gamma = \lim_{z\to 0} \left\{\Gamma(z) - \frac1{z} \right\}
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| = \lim_{z\to 0} \left\{\Psi(z) + \frac1{z} \right\}.</math>
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| Further limit results are (Krämer, 2005):
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| :<math> \lim_{z\to 0} \frac1{z}\left\{\frac1{\Gamma(1+z)} - \frac1{\Gamma(1-z)} \right\} = 2\gamma</math>
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| :<math> \lim_{z\to 0} \frac1{z}\left\{\frac1{\Psi(1-z)} - \frac1{\Psi(1+z)} \right\} = \frac{\pi^2}{3\gamma^2}.</math>
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| A limit related to the [[beta function]] (expressed in terms of [[gamma function]]s) is
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| :<math> \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\} </math>
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| :<math>\gamma = \lim\limits_{m \to \infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\ln(\Gamma(k+1)).</math>
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| ===Relation to the zeta function===
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| <math>\gamma</math> can also be expressed as an [[series (mathematics)|infinite sum]] whose terms involve the [[Riemann zeta function]] evaluated at positive integers:
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| :<math>\begin{align}\gamma &= \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m} \\
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| &= \ln \left ( \frac{4}{\pi} \right ) + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align} </math>
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| Other series related to the zeta function include:
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| :<math>\begin{align} \gamma &= \frac{3}{2}- \ln 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m} [\zeta(m)-1] \\
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| &= \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 ] \\
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| &= \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} </math>
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| 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.
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| Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit (Sondow, 1998)
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| :<math> \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 ) = \lim_{s \to 0} \frac{\zeta(1+s)+\zeta(1-s)}{2}</math>
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| and [[Charles Jean de la Vallée-Poussin|de la Vallée-Poussin's]] formula
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| :<math>\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}</math>
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| 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:
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| :<math>\gamma = \sum_{k=1}^n \frac{1}{k} - \ln n -
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| \sum_{m=2}^\infty \frac{\zeta (m,n+1)}{m}</math>
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| where ζ(''s'',''k'') is the [[Hurwitz zeta function]]. The sum in this equation involves the [[harmonic number]]s, ''H''<sub>''n''</sub>. Expanding some of the terms in the Hurwitz zeta function gives:
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| :<math>
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| H_n = \ln n + \gamma + \frac {1} {2n} - \frac {1} {12n^2} + \frac {1} {120n^4} - \varepsilon </math>, where <math>0 < \varepsilon < \frac {1} {252n^6}.</math>
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| ===Integrals===
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| <math>\gamma</math> equals the value of a number of definite [[integral]]s:
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| :<math>\begin{align}\gamma &= - \int_0^\infty { e^{-x} \ln x }\,dx = -4\int_0^\infty { e^{-x^2} x \ln x }\,dx\\
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| &= -\int_0^1 \ln\ln\left (\frac{1}{x}\right) dx \\
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| &= \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\\
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| &= \int_0^\infty \left (\frac1{1+x^k}-e^{-x} \right)\frac{dx}{x},\quad k>0\\
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| &= \int_0^1 H_{x} dx \end{align} </math>
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| where <math>H_{x}</math> is the [[Harmonic number|fractional Harmonic number]].
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| Definite integrals in which <math>\gamma</math> appears include:
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| :<math> \int_0^\infty { e^{-x^2} \ln x }\,dx = -\tfrac14(\gamma+2 \ln 2) \sqrt{\pi} </math>
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| :<math> \int_0^\infty { e^{-x} \ln^2 x }\,dx = \gamma^2 + \frac{\pi^2}{6} .</math>
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| One can express <math>\gamma</math> using a special case of [[Hadjicostas's formula]] as a [[double integral]] (Sondow 2003a, 2005) with equivalent series:
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| : <math> \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 ).
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| </math>
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| An interesting comparison by J. Sondow (2005) is the double integral and alternating series
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| :<math> \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). </math>
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| It shows that <math>\ln \left ( \frac{4}{\pi} \right )</math> may be thought of as an "alternating Euler constant".
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| The two constants are also related by the pair of series (see Sondow 2005 #2)
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| :<math> \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} = \gamma </math>
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| :<math> \sum_{n=1}^\infty \frac{N_1(n) - N_0(n)}{2n(2n+1)} = \ln \left ( \frac{4}{\pi} \right ) </math>
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| where ''N''<sub>1</sub>(''n'') and ''N''<sub>0</sub>(''n'') are the number of 1's and 0's, respectively, in the [[base 2]] expansion of ''n''.
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| We have also Catalan's 1875 integral (see Sondow and Zudilin)
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| :<math> \gamma = \int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx. </math>
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| ===Series expansions===
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| Euler showed that the following [[infinite series]] approaches <math> \gamma </math>:
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| :<math>\gamma = \sum_{k=1}^\infty \left[ \frac{1}{k} - \ln \left( 1 + \frac{1}{k} \right) \right].</math>
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| The series for <math>\gamma</math> is equivalent to series Nielsen found in 1897:
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| :<math> \gamma = 1 - \sum_{k=2}^{\infty}(-1)^k\frac{\lfloor\log_2 k\rfloor}{k+1}.</math>
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| In 1910, Vacca found the closely related series:
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| :<math>{
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| \gamma = \sum_{k=2}^\infty (-1)^k \frac{ \left \lfloor \log_2 k \right \rfloor}{k}
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| = \frac12-\frac13
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| + 2\left(\frac14 - \frac15 + \frac16 - \frac17\right)
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| + 3\left(\frac18 - \frac19 + \frac1{10} - \frac1{11} + \dots - \frac1{15}\right) + \dots
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| }</math>
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| where <math>\log_2</math> is the [[logarithm]] of base 2 and <math> \lfloor \, \rfloor </math> is the [[floor function]].
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| In 1926 he found a second series:
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| :<math>{\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}.</math>
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| From the [[Ernst Eduard Kummer|Kummer]]-expansion of the gamma function we get:
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| :<math> \gamma = \ln\pi - 4\ln\Gamma(\tfrac34) + \frac4{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\ln(2k+1)}{2k+1}.</math>
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| Series of prime numbers:
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| :<math>\begin{align} \gamma = \lim_{n \to \infty} \left( \ln n - \sum_{p \le n} \frac{ \ln p }{ p-1 } \right)\end{align}.</math> <ref>http://mathworld.wolfram.com/MertensConstant.html (15)</ref>
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| ===Asymptotic expansions===
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| <math>\gamma</math> equals the following asymptotic formulas (where <math>H_n</math> is the ''n''th [[harmonic number]].)
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| :<math>\gamma \sim H_n - \ln \left( n \right) - \frac{1}{{2n}} + \frac{1}{{12n^2 }} - \frac{1}{{120n^4 }} + ...</math>
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| :(''Euler'')
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| :<math>\gamma \sim H_n - \ln \left( {n + \frac{1}{2} + \frac{1}{{24n}} - \frac{1}{{48n^3 }} + ...} \right)</math>
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| :(''Negoi'')
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| :<math>\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 }} - ...</math>
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| :(''Cesaro'')
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| The third formula is also called the Ramanujan expansion. | |
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| ===Relations with the reciprocal logarithm===
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| The reciprocal logarithm function (Krämer, 2005)
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| :<math>\frac{z}{\ln(1-z)} = \sum_{n=0}^{\infty}C_nz^n, \quad |z|<1,</math>
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| has a deep connection with Euler's constant and was studied by [[James Gregory (mathematician)|James Gregory]] in connection with [[numerical integration]]. The coefficients <math>C_n</math> are called [[Gregory coefficients]]; the first six were given in a letter to [[John Collins (mathematician)|John Collins]] in 1670. From the equations
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| :<math>C_0 = -1\;,\quad \sum_{k=0}^n\frac{C_k}{n+1-k} = 0,\quad n=1,2,3,\dots</math>
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| , which can be used [[recursively]] to get these coefficients for all <math>n \ge 1</math>, we get the table
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| {| class="wikitable" border="1" align="center"
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| |-
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| ! n
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| ! width="40" |1
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| ! width="40" |2
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| ! width="50" |3
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| ! width="50" |4
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| ! width="50" |5
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| ! width="50" |6
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| ! width="50" |7
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| ! width="50" |8
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| ! width="50" |9
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| ! width="80" |10
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| !| [[OEIS]] sequences
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| |- align="center"
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| ! C<sub>''n''</sub>
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| | <math>\tfrac12</math>
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| | <math>\tfrac1{12}</math>
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| | <math>\tfrac1{24}</math>
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| | <math>\tfrac{19}{720}</math>
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| | <math>\tfrac3{160}</math>
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| | <math>\tfrac{863}{60480}</math>
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| | <math>\tfrac{275}{24192}</math>
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| | <math>\tfrac{33953}{3628800}</math>
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| | <math>\tfrac{8183}{1036800}</math>
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| | <math>\tfrac{3250433}{479001600}</math>
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| | {{OEIS2C|id=A002206}} (numerators),
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| {{OEIS2C|id=A002207}}(denominators)
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| |}
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| Gregory coefficients are similar to [[Bernoulli numbers]] and satisfy the asymptotic relation
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| :<math>C_n = \frac1{n\ln^2 n} - \mathcal{O}\left(\frac1{n\ln^3 n}\right),\quad n\to\infty,</math>
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| and the integral representation
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| :<math>C_n = \int_0^{\infty}\frac{dx}{(1+x)^n\left(\ln^2 x + \pi^2\right)},\quad n=1,2,\dots.</math>
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| Euler's constant has the integral representations
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| :<math>\gamma = \int_0^{\infty}\frac{\ln(1+x)}{\ln^2 x + \pi^2}\cdot\frac{dx}{x^2}
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| = \int_{-\infty}^{\infty}\frac{\ln(1+e^{-x})}{x^2 + \pi^2}\,e^x\,dx.</math>
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| A very important expansion of [[Gregorio Fontana]] (1780) is:
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| :<math>
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| \begin{align}
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| H_n &= \gamma + \log n + \frac1{2n}
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| - \sum_{k=2}^{\infty}\frac{(k-1)!C_k}{n(n+1)\dots(n+k-1)},\quad n=1,2,\dots,\\
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| &= \gamma + \log n + \frac1{2n}
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| - \frac1{12n(n+1)} - \frac1{12n(n+1)(n+2)} - \frac{19}{120n(n+1)(n+2)(n+3)} - \dots
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| \end{align}
| |
| </math>
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| which is convergent for all ''n''.
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| Weighted sums of the Gregory coefficients give different constants:
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| :<math>
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| \begin{align}
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| 1 &= \sum_{n=1}^{\infty}C_n
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| = \tfrac12 + \tfrac1{12} + \tfrac1{24} + \tfrac{19}{720} + \tfrac3{160} + \dots,\\
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| \frac1{\log2} - 1 &= \sum_{n=1}^{\infty}(-1)^{n+1}C_n
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| = \tfrac12 - \tfrac1{12} + \tfrac1{24} - \tfrac{19}{720} + \tfrac3{160} - \dots,\\
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| \gamma &= \sum_{n=1}^{\infty}\frac{C_n}{n}
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| = \tfrac12 + \tfrac1{24} + \tfrac1{72} + \tfrac{19}{2880} + \tfrac3{800} + \dots.
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| \end{align}
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| </math>
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| | |
| ===''e''<sup>γ</sup>===
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| The constant ''e''<sup>γ</sup> is important in number theory. Some authors denote this quantity simply as <math> \gamma^\prime </math>. ''e''<sup>γ</sup> equals the following [[limit of a sequence|limit]], where ''p''<sub>''n''</sub> is the ''n''th [[prime number]]:
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| :<math>e^\gamma = \lim_{n \to \infty} \frac {1} {\ln p_n} \prod_{i=1}^n \frac {p_i} {p_i - 1}.</math>
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| This restates the third of [[Mertens' theorems]]. The numerical value of ''e''<sup>γ</sup> is:
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| :1.78107241799019798523650410310717954916964521430343 … {{OEIS2C|id=A073004}}.
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| Other [[infinite product]]s relating to ''e''<sup>γ</sup> include:
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| :<math> \frac{e^{1+\gamma /2}}{\sqrt{2\,\pi}} = \prod_{n=1}^\infty e^{-1+1/(2\,n)}\,\left (1+\frac{1}{n} \right )^n </math>
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| | |
| :<math> \frac{e^{3+2\gamma}}{2\, \pi} = \prod_{n=1}^\infty e^{-2+2/n}\,\left (1+\frac{2}{n} \right )^n. </math>
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| | |
| These products result from the [[Barnes G-function]].
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| We also have
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| :<math> 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}
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| \left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/5} \cdots </math>
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| where the ''n''th factor is the (''n''+1)st root of
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| :<math>\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}.</math>
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| This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using [[hypergeometric function]]s.
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| ===Continued fraction===
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| The [[continued fraction]] expansion of <math> \gamma </math> is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] {{OEIS2C|id=A002852}}, of which there is no ''apparent'' pattern. The continued fraction has '''at least''' 470,000 terms,<ref name=Havil>Havil 2003 p 97.</ref> and it has infinitely many terms [[if and only if]] <math>\gamma</math> is irrational.
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| ==Generalizations==
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| ''Euler's generalized constants'' are given by
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| :<math>\gamma_\alpha = \lim_{n \to \infty} \left[ \sum_{k=1}^n \frac{1}{k^\alpha} - \int_1^n \frac{1}{x^\alpha} \, dx \right],</math>
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| | |
| for 0 < α < 1, with <math>\gamma</math> as the special case α = 1.<ref>Havil, 117-118</ref> This can be further generalized to
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| :<math>c_f = \lim_{n \to \infty} \left[ \sum_{k=1}^n f(k) - \int_1^n f(x) \, dx \right]</math>
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| | |
| for some arbitrary decreasing function ''f''. For example,
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| :<math>f_n(x) = \frac{\ln^n x}{x}</math>
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| | |
| gives rise to the [[Stieltjes constants]], and
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| :<math>f_a(x) = x^{-a}</math>
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| gives
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| :<math>\gamma_{f_a} = \frac{(a-1)\zeta(a)-1}{a-1}</math>
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| | |
| where again the limit
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| :<math>\gamma = \lim_{a\to1}\left[ \zeta(a) - \frac{1}{a-1}\right]</math> | |
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| appears.
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| 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<ref>{{cite journal|first1=M. | last1=Ram Murty | first2=N. | last2=Saradha | title=Euler-lehmer constants and a conjecture of Erdos| journal = JNT| doi=10.1016/j.jnt.2010.07.004| year=2010|volume=130|pages=2671–2681}}</ref>
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| ,<ref>{{cite journal| first1=D. H. | last1=Lehmer|year=1975|title=Euler constants for arithmetical progressions | journal=Acta Arithm. |volume=27 |number=1| pages=125–142|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf}}</ref> | |
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| :<math>\gamma(a,q) = \lim_{x\to \infty}\left ( \sum_{0<n\le x \atop n\equiv a \pmod q} \frac{1}{n}-\frac{\log x}{q} \right ).</math>
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| The basic properties are
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| :<math>\gamma(0,q) = \frac{\gamma -\log q}{q},</math>
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| :<math>\sum_{a=0}^{q-1} \gamma(a,q)=\gamma,</math>
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| :<math>q\gamma(a,q) = \gamma-\sum_{j=1}^{q-1}e^{-2\pi aij/q}\log(1-e^{2\pi ij/q}),</math>
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| | |
| and if {{math|gcd(a,q){{=}}d}} then
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| :<math>q\gamma(a,q) = \frac{q}{d}\gamma(a/d,q/d)-\log d.</math>
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| ==Published digits==
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| 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 ...'''181'''12090082'''39''' when the correct value is ...'''065'''12090082'''40'''.
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| | |
| {| class="wikitable" style="margin: 1em auto 1em auto"
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| |+ '''Published Decimal Expansions of ''<math>\gamma</math>'' '''
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| ! Date || Decimal digits || Author
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| |-
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| | 1734 || 5 || [[Leonhard Euler]]
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| |-
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| | 1735 || 15 || Leonhard Euler
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| |-
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| | 1790 || 19 || [[Lorenzo Mascheroni]]
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| |-
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| | 1809 || 22 || [[Johann Georg von Soldner|Johann G. von Soldner]]
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| |-
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| | 1811 || 22 || [[Carl Friedrich Gauss]]
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| |-
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| | 1812 || 40 || [[Friedrich Bernhard Gottfried Nicolai]]
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| |-
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| | 1857 || 34 || Christian Fredrik Lindman
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| |-
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| | 1861 || 41 || Ludwig Oettinger
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| |-
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| | 1867 || 49 || [[William Shanks]]
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| |-
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| | 1871 || 99 || [[James Whitbread Lee Glaisher|James W.L. Glaisher]]
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| |-
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| | 1871 || 101 || William Shanks
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| |-
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| | 1877 || 262 || [[John Couch Adams|J. C. Adams]]
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| |-
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| | 1952 || 328 || [[John Wrench|John William Wrench, Jr.]]
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| |-
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| | 1961 || 1050 || Helmut Fischer and Karl Zeller
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| |-
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| | 1962 || 1,271 || [[Donald Knuth]]
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| |-
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| | 1962 || 3,566 || Dura W. Sweeney
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| |-
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| | 1973 || 4,879 || William A. Beyer and [[Michael S. Waterman]]
| |
| |-
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| | 1977 || 20,700 || [[Richard Brent (scientist)|Richard P. Brent]]
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| |-
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| | 1980 || 30,100 || Richard P. Brent & [[Edwin McMillan|Edwin M. McMillan]]
| |
| |-
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| | 1993 || 172,000 || [[Jonathan Borwein]]
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| |-
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| | 2009 || 29,844,489,545 || Alexander J. Yee & Raymond Chan<ref>[http://www.numberworld.org/nagisa_runs/computations.html Nagisa – Large Computations]</ref>
| |
| |}
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| ==See also==
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| {{portal|Mathematics}}
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| {{quote box
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| |align=left
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| |quote={{Irrational numbers}}
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| |fontsize=100%
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| }}
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| {{-}}
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| ==Notes==
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| ;Footnotes
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| {{Reflist}}
| |
| | |
| ;References
| |
| *{{cite journal|author=Borwein, Jonathan M., David M. Bradley, Richard E. Crandall
| |
| |title=Computational Strategies for the Riemann Zeta Function
| |
| |journal=Journal of Computational and Applied Mathematics
| |
| |year=2000
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| |volume=121
| |
| |pages=11
| |
| |url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf}} Derives γ as sums over Riemann zeta functions.
| |
| *Gourdon, Xavier, and Sebah, P. (2002) "[http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Collection of formulas for Euler's constant, γ.]"
| |
| *Gourdon, Xavier, and Sebah, P. (2004) "[http://numbers.computation.free.fr/Constants/Gamma/gamma.html The Euler constant: γ.]"
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| *[[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.
| |
| *[[Jonathan Sondow|Sondow, Jonathan]] (1998) "[http://home.earthlink.net/~jsondow/id8.html An antisymmetric formula for Euler's constant,]" ''[[Mathematics Magazine]] 71'': 219-220.
| |
| *[[Jonathan Sondow|Sondow, Jonathan]] (2002) "[http://arXiv.org/abs/math.NT/0211075 A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant.]" With an Appendix by [http://wain.mi.ras.ru/zlobin/ Sergey Zlobin], Mathematica Slovaca 59'': 307-314.
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| *{{cite arxiv | first1=Jonathan | last1= Sondow | year=2003 | eprint=math.CA/0306008 | title= An infinite product for e<sup>γ</sup> via hypergeometric formulas for Euler's constant, γ}}
| |
| *[[Jonathan Sondow|Sondow, Jonathan]] (2003a) "[http://arXiv.org/abs/math.NT/0209070 Criteria for irrationality of Euler's constant,]" ''[[Proceedings of the American Mathematical Society]] 131'': 3335-3344.
| |
| *[[Jonathan Sondow|Sondow, Jonathan]] (2005) "[http://arXiv.org/abs/math.CA/0211148 Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula,]" ''[[American Mathematical Monthly]] 112'': 61-65.
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| *[[Jonathan Sondow|Sondow, Jonathan]] (2005) [http://arXiv.org/abs/math.NT/0508042 "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π.]"
| |
| * {{cite arxiv| first1=Jonathan |last1=Sondow | first2=Wadim | last2=Zudilin | year=2006 |eprint=math.NT/0304021 |title= 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
| |
| *{{cite book
| |
| |first = Julian
| |
| |last = Havil
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| |year = 2003
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| |title = Gamma: Exploring Euler's Constant
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| |publisher = Princeton University Press
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| |isbn = 0-691-09983-9
| |
| }}
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| *{{ cite journal| first1=E. A. |last1= Karatsuba |title=Fast evaluation of transcendental functions | journal=Probl. Inf. Transm. |volume =27 | number=44 | pages=339–360 |year=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)
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| *M. Lerch, Expressions nouvelles de la constante d'Euler. Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften 42, 5 p. (1897)
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| * {{cite arxiv| first1=Jeffrey C|last1= Lagarias |title=Euler's constant: Euler's work and modern developments | eprint=1303.1856}}, [[Bulletin of the American Mathematical Society]] 50 (4): 527-628 (2013)
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| ==External links==
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| *{{mathworld|urlname=Euler-MascheroniConstant|title=Euler-Mascheroni constant}}
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| *Krämer, Stefan "[http://www.math.uni-goettingen.de/skraemer/gamma.html Euler's Constant γ=0.577... Its Mathematics and History.]"
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| *[http://home.earthlink.net/~jsondow/ Jonathan Sondow.]
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| *[http://www.ccas.ru/personal/karatsuba/algen.htm Fast Algorithms and the FEE Method], E.A. Karatsuba (2005)
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| {{DEFAULTSORT:Euler-Mascheroni Constant}}
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| [[Category:Mathematical constants]]
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| [[Category:Unsolved problems in mathematics]]
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