View source for Euler–Mascheroni constant

{{redirect|Euler's constant|the base of the natural logarithm, {{mvar|e}} ≈ 2.718...|e (mathematical constant){{!}}{{mvar|e}} (mathematical constant)}}
{{Use Harvard referencing|date=July 2019}}
[[File:gamma-area.svg|thumb|The area of the blue region converges to the Euler–Mascheroni constant.]]

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 [[gamma (letter)|gamma]] ({{mvar|γ}}).

It is defined as the [[limit of a sequence|limiting]] difference between the [[harmonic series (mathematics)|harmonic series]] and the [[natural logarithm]]:

:<math>\begin{align}
\gamma &= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)\\[5px]
&=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.
\end{align}</math>

Here, <math>\lfloor x\rfloor</math> represents the [[floor and ceiling functions|floor function]].

The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:
: {{gaps|0.57721|56649|01532|86060|65120|90082|40243|10421|59335|93992|...}} {{nowrap|{{OEIS|A001620}}}}

{| class="infobox bordered" cellpadding=5
|-
|[[Binary numeral system|Binary]]
|{{gaps|0.1001|0011|1100|0100|0110|0111|1110|0011|0111|1101|...}}
|-
|[[Decimal]]
|{{gaps|0.57721|56649|01532|86060|65120|90082|40243|10421|...}}
|-
|[[Hexadecimal]]
|{{gaps|0.93C4|67E3|7DB0|C7A4|D1BE|3F81|0152|CB56|A1CE|CC3A|...}}
|-
|[[Continued fraction]]
|{{nowrap|[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, ...]}}<br /><small>(It is not known whether this continued fraction is [[finite continued fraction|finite]], infinite [[periodic continued fraction|periodic]] or infinite non-periodic. <br>Shown in [[continued fraction#Notations for continued fractions|linear notation]])</small><br />Source: {{harvnb|Sloane}}
|}

==History==
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 {{math|''C''}} and {{math|''O''}} for the constant. In 1790, [[Italy|Italian]] mathematician [[Lorenzo Mascheroni]] used the notations {{math|''A''}} and {{math|''a''}} for the constant. The notation {{mvar|γ}} 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]] {{harv|Lagarias|2013}}.  For example, the [[Germany|German]] mathematician [[Carl Anton Bretschneider]] used the notation {{mvar|γ}} in 1835 {{harv|Bretschneider|1837|loc="{{mvar|γ}} = {{math|''c''}} = {{nowrap|0,577215 664901 532860 618112 090082 3..}}" on [https://books.google.com/books?hl=fi&id=OAoPAAAAIAAJ&pg=PA260 p. 260]}} and [[Augustus De Morgan]] used it in a textbook published in parts from 1836 to 1842 {{harv|De Morgan|1836–1842|loc="{{mvar|γ}}" on [https://books.google.com/books?id=95x4IrIcHrgC&pg=PA578 p. 578]}}

==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 [[Laurent 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]]
* In [[dimensional regularization]] of [[Feynman diagram]]s in [[quantum field theory]]
* The calculation of the [[Meissel–Mertens constant]]
* The third of [[Mertens' theorems]]*
* Solution of the second kind to [[Bessel's equation]]
* In the regularization/[[renormalization]] of the [[harmonic series (mathematics)|harmonic series]] as a finite value
* The [[mean]] of the [[Gumbel distribution]]
* 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.
* The answer to the [[coupon collector's problem]]*
* In some formulations of [[Zipf's law]]
* A definition of the [[trigonometric integral#Cosine integral|cosine integral]]*
* Lower bounds to a [[Prime gap#Lower bounds|prime gap]]
* An upper bound on [[Shannon entropy]] in [[quantum information science|quantum information theory]] {{harv|Caves|Fuchs|1996}}

==Properties==
The number {{mvar|γ}} has not been proved [[algebraic number|algebraic]] or [[transcendental number|transcendental]]. In fact, it is not even known whether {{mvar|γ}} is [[irrational number|irrational]]. [[Continued fraction]] analysis reveals that if {{mvar|γ}} is [[rational number|rational]], its denominator must be greater than 10<sup>242080</sup> {{harv|Havil|2003|p=97}}. The ubiquity of {{mvar|γ}} revealed by the large number of equations below makes the irrationality of {{mvar|γ}} a major open question in mathematics. Also see {{Harv|Sondow|2003a}}.

===Relation to gamma function===
{{mvar|γ}} is related to the [[digamma function]] {{math|Ψ}}, and hence the [[derivative]] of the [[gamma function]] {{math|Γ}}, when both functions are evaluated at 1. Thus:
:<math>-\gamma = \Gamma'(1) = \Psi(1). </math>

This is equal to the limits:
:<math>\begin{align}-\gamma &= \lim_{z\to 0}\left(\Gamma(z) - \frac1{z}\right) \\&= \lim_{z\to 0}\left(\Psi(z) + \frac1{z}\right).\end{align}</math>

Further limit results are {{Harv|Krämer|2005}}:

:<math>\begin{align} \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}. \end{align}</math>

A limit related to the [[beta function]] (expressed in terms of [[gamma function]]s) is
:<math>\begin{align} \gamma &= \lim_{n\to\infty}\left(\frac{ \Gamma\left(\frac1{n}\right) \Gamma(n+1)\, n^{1+\frac1{n}}}{\Gamma\left(2+n+\frac1{n}\right)} - \frac{n^2}{n+1}\right) \\
&= \lim\limits_{m\to\infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\ln\big(\Gamma(k+1)\big). \end{align}</math>

===Relation to the zeta function===
{{mvar|γ}} can also be expressed as an [[series (mathematics)|infinite sum]] whose terms involve the [[Riemann zeta function]] evaluated at positive integers:
:<math>\begin{align}\gamma &= \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m} \\
 &= \ln\frac4{\pi} + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align} </math>

Other series related to the zeta function include:
:<math>\begin{align} \gamma &= \tfrac3{2}- \ln 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m}\big(\zeta(m)-1\big) \\
 &= \lim_{n\to\infty}\left(\frac{2n-1}{2n} - \ln n + \sum_{k=2}^n \left(\frac1{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^{mn}}{(m+1)!} \sum_{t=0}^m \frac1{t+1} - n \ln 2+ O \left (\frac1{2^{n}\, e^{2^n}}\right)\right).\end{align}</math>

The error term in the last equation is a rapidly decreasing function of {{math|''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 {{harv|Sondow|1998}}:
:<math>\begin{align} \gamma &= \lim_{s\to 1^+}\sum_{n=1}^\infty \left(\frac1{n^s}-\frac1{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} \end{align}</math>

and [[Charles Jean de la Vallée-Poussin|de la Vallée-Poussin's]] formula
:<math>\gamma = \lim_{n\to\infty}\frac1{n}\, \sum_{k=1}^n \left(\left\lceil \frac{n}{k} \right\rceil - \frac{n}{k}\right)</math>
where <math>\lceil\, \rceil</math> are [[ceiling function|ceiling]] brackets.

Closely related to this is the [[rational zeta series]] expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:

:<math>\gamma = \sum_{k=1}^n \frac1{k} - \ln n -\sum_{m=2}^\infty \frac{\zeta(m,n+1)}{m},</math>

where {{math|''ζ''(''s'',''k'')}} is the [[Hurwitz zeta function]]. The sum in this equation involves the [[harmonic number]]s, {{math|''H''<sub>''n''</sub>}}. Expanding some of the terms in the Hurwitz zeta function gives:
:<math>H_n = \ln(n) + \gamma + \frac1{2n} - \frac1{12n^2} + \frac1{120n^4} - \varepsilon,</math>
where {{math|0 < ''ε'' < {{sfrac|1|252''n''<sup>6</sup>}}.}}

{{mvar|γ}} can also be expressed as follows where {{math|''A''}} is the [[Glaisher–Kinkelin constant]]:

:<math>\gamma =12\,\log(A)-\log(2\,\pi)+\frac{6}{\pi^2}\,\zeta'(2)</math>

{{mvar|γ}} can also be expressed as follows, which can be proven by expressing the [[Riemann zeta function|zeta function]] as a [[Laurent series]]:

:<math>\gamma=\lim_{n\to\infty}\biggl(-n+\zeta\Bigl(\frac{n+1}{n}\Bigr)\biggr)</math>

===Integrals===
{{mvar|γ}} equals the value of a number of definite [[integral]]s:
:<math>\begin{align}\gamma &= - \int_0^\infty e^{-x} \ln x \,dx \\
 &= -\int_0^1 \ln\left(\ln\frac 1 x \right) dx \\
 &= \int_0^\infty \left(\frac1{e^x-1}-\frac1{x\cdot e^x} \right)dx \\
 &= \int_0^1\left(\frac1{\ln x} + \frac1{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} </math>
where {{math|''H''<sub>''x''</sub>}} is the [[Harmonic number#Harmonic numbers for real and complex values|fractional harmonic number]].

Definite integrals in which {{mvar|γ}} appears include:
:<math>\begin{align}\int_0^\infty e^{-x^2} \ln x \,dx &= -\frac{(\gamma+2\ln 2)\sqrt{\pi}}{4} \\
\int_0^\infty e^{-x} \ln^2 x \,dx &= \gamma^2 + \frac{\pi^2}{6} . \end{align}</math>

One can express {{mvar|γ}} using a special case of [[Hadjicostas's formula]] as a [[Multiple integral#Double integral|double integral]] {{Harv|Sondow|2003a}} and {{Harv|Sondow|2005}} with equivalent series:
:<math>\begin{align}\gamma &= \int_0^1 \int_0^1 \frac{x-1}{(1-xy)\ln xy}\,dx\,dy \\&= \sum_{n=1}^\infty \left(\frac 1 n -\ln\frac{n+1} n \right).\end{align}</math>

An interesting comparison by {{Harv|Sondow|2005}} is the double integral and alternating series
:<math>\begin{align} \ln\frac 4 \pi &= \int_0^1 \int_0^1 \frac{x-1}{(1+xy)\ln xy} \,dx\,dy \\&= \sum_{n=1}^\infty \left((-1)^{n-1}\left(\frac 1 n -\ln\frac{n+1} n \right)\right).\end{align}</math>

It shows that {{math|ln {{sfrac|4|π}}}} may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series {{Harv|Sondow|2005a}}
:<math>\begin{align} \gamma &= \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} \\
\ln\frac4{\pi} &= \sum_{n=1}^\infty \frac{N_1(n) - N_0(n)}{2n(2n+1)} ,\end{align}</math>
where {{math|''N''<sub>1</sub>(''n'')}} and {{math|''N''<sub>0</sub>(''n'')}} are the number of 1s and 0s, respectively, in the [[Binary number|base 2]] expansion of {{math|''n''}}.

We have also [[Eugène Charles Catalan|Catalan]]'s 1875 integral (see {{harvnb|Sondow|Zudilin|2006}})
:<math>\gamma = \int_0^1 \left(\frac1{1+x}\sum_{n=1}^\infty x^{2^n-1}\right)\,dx.</math>

===Series expansions===
In general, 
:<math>
\gamma = \lim_{n \to \infty}\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} + \ldots + \frac{1}{n} - \ln(n+\alpha) \right) \equiv \lim_{n \to \infty} \gamma_n(\alpha)
</math>
for any <math>\alpha > -n</math>. However, the rate of convergence of this expansion depends significantly on <math>\alpha</math>. In particular, <math>\gamma_n(1/2)</math> exhibits much more rapid convergence than the conventional expansion <math>\gamma_n(0)</math> ({{harvnb|DeTemple|1993}}; {{harvnb|Havil|2003|pp=75–78}}). This is because

:<math>
\frac{1}{2(n+1)} < \gamma_n(0) - \gamma < \frac{1}{2n},
</math>

while

:<math>
\frac{1}{24(n+1)^2} < \gamma_n(1/2) - \gamma < \frac{1}{24n^2}.
</math>

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the following [[infinite series]] approaches {{mvar|γ}}:
:<math>\gamma = \sum_{k=1}^\infty \left(\frac 1 k - \ln\left(1+\frac 1 k \right)\right).</math>

The series for {{mvar|γ}} is equivalent to a series [[Niels Nielsen (mathematician)|Nielsen]] found in 1897 ({{harvnb|Krämer|2005}}, {{harvnb|Blagouchine|2016}}):
:<math>\gamma = 1 - \sum_{k=2}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor}{k+1}.</math>

In 1910, [[Giovanni Vacca (mathematician)|Vacca]] found the closely related series ({{harvnb|Vacca|1910}},{{Citation not found}} {{harvnb|Glaisher|1910}}, {{harvnb|Hardy|1912}}, {{harvnb|Vacca|1925}},{{Citation not found}} {{harvnb|Kluyver|1927}}, {{harvnb|Krämer|2005}}, {{harvnb|Blagouchine|2016}})
:<math>\begin{align}
\gamma & = \sum_{k=2}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor} k \\[5pt]
& = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1{10} - \tfrac1{11} + \cdots - \tfrac1{15}\right) + \cdots,
\end{align}</math>

where {{math|log<sub>2</sub>}} is the [[binary logarithm|logarithm to base 2]] and {{math|⌊ ⌋}} is the [[Floor and ceiling functions|floor function]].

In 1926 he found a second series:
:<math>\begin{align}
\gamma + \zeta(2) & = \sum_{k=2}^\infty \left( \frac1{\left\lfloor\sqrt{k}\right\rfloor^2} - \frac1{k}\right) \\[5pt]
& = \sum_{k=2}^\infty \frac{k - \left\lfloor\sqrt{k}\right\rfloor^2}{k \left\lfloor \sqrt{k} \right\rfloor^2} \\[5pt]
&= \frac12 + \frac23 + \frac1{2^2}\sum_{k=1}^{2\cdot 2} \frac{k}{k+2^2} + \frac1{3^2}\sum_{k=1}^{3\cdot 2} \frac{k}{k+3^2} + \cdots
\end{align}</math>

From the [[Carl Johan Malmsten|Malmsten]]–[[Ernst Kummer|Kummer]] expansion for the logarithm of the gamma function {{harv|Blagouchine|2014}} we get:
:<math>\gamma = \ln\pi - 4\ln\left(\Gamma(\tfrac34)\right) + \frac 4 \pi \sum_{k=1}^\infty (-1)^{k+1}\frac{\ln(2k+1)}{2k+1}.</math>

An important expansion for Euler's constant is due to [[Gregorio Fontana|Fontana]] and [[Lorenzo Mascheroni|Mascheroni]]

:<math>\gamma = \sum_{n=1}^\infty \frac{|G_n|}{n} = \frac12 + \frac1{24} + \frac1{72} + \frac{19}{2880} + \frac3{800} + \cdots,</math>
where {{math|''G<sub>n</sub>''}} are [[Gregory coefficients]] ({{harvnb|Krämer|2005}}, {{harnvb|Blagouchine|2016}}, {{harvnb|Blagouchine|2018}}) This series is the special case <math>k=1</math> of the expansions
:<math>\begin{align}
 \gamma &= H_{k-1}  - \ln k + \sum_{n=1}^{\infty}\frac{(n-1)!|G_n|}{k(k+1) \cdots (k+n-1)} && \\
     &= H_{k-1} - \ln k + \frac1{2} + \frac1{12k(k+1)} + \frac1{12k(k+1)(k+2)} + \frac{19}{120k(k+1)(k+2)(k+3)} + \cdots &&
\end{align}</math>
convergent for <math>k=1,2,\ldots</math>

A similar series with the Cauchy numbers of the second kind {{math|''C<sub>n</sub>''}} is ({{harvnb|Blagouchine|2016}}; {{harvnb|Alabdulmohsin|2018|pp=147–148}}) 
:<math>\gamma = 1 -  \sum_{n=1}^\infty \frac{C_{n}}{n \, (n+1)!} =1- \frac{1}{4} -\frac{5}{72} - \frac{1}{32} - \frac{251}{14400} - \frac{19}{1728} - \ldots</math>

Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series
:<math>\gamma=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}\Big\{\psi_{n}(a)+ \psi_{n}\Big(-\frac{a}{1+a}\Big)\Big\},
\quad a>-1</math>
where {{math|''ψ<sub>n</sub>''(''a'')}} are the ''[[Bernoulli polynomials of the second kind]]'', which are defined by the generating function
: <math>
\frac{z(1+z)^s}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(s) ,\qquad |z|<1,
</math>
For any rational {{math|''a''}} this series contains rational terms only. For example, at {{math|''a'' {{=}} 1}}, it becomes
: <math>\gamma=\frac{3}{4} - \frac{11}{96} - \frac{1}{72} - \frac{311}{46080} - \frac{5}{1152} - \frac{7291}{2322432}  - \frac{243}{100352} - \ldots</math>
see {{OEIS2C|id=A302120}} and {{OEIS2C|id=A302121}}. Other series with the same polynomials include these examples:
: <math>\gamma= -\ln(a+1) - \sum_{n=1}^\infty\frac{(-1)^n \psi_{n}(a)}{n},\qquad \Re(a)>-1 </math>
and
: <math>\gamma= -\frac{2}{1+2a} \left\{\ln\Gamma(a+1) -\frac{1}{2}\ln(2\pi) + \frac{1}{2} + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{n}\right\},\qquad \Re(a)>-1 </math>
where {{math|Γ(''a'')}} is the [[gamma function]] {{harv|Blagouchine|2018}}.

A series related to the Akiyama-Tanigawa algorithm is
: <math>\gamma= \ln(2\pi) - 2 - 2 \sum_{n=1}^\infty\frac{(-1)^n G_{n}(2)}{n}=
\ln(2\pi) - 2 + \frac{2}{3} + \frac{1}{24}+ \frac{7}{540} + \frac{17}{2880}+ \frac{41}{12600} + \ldots  </math>
where {{math|''G''<sub>''n''</sub>(2)}} are the [[Gregory coefficients]] of the second order {{harv|Blagouchine|2018}}.
  
Series of [[prime number]]s:
:<math>\gamma = \lim_{n\to\infty}\left(\ln n - \sum_{p\le n}\frac{\ln p}{p-1}\right).</math>

===Asymptotic expansions===
{{mvar|γ}} equals the following asymptotic formulas (where {{math|''H<sub>n</sub>''}} is the {{math|''n''}}th [[harmonic number]]):
:<math>\gamma \sim H_n - \ln n - \frac1{2n} + \frac1{12n^2} - \frac1{120n^4} + \cdots</math> (''Euler'')
:<math>\gamma \sim H_n - \ln\left({n + \frac1{2} + \frac1{24n} - \frac1{48n^3} + \cdots}\right)</math> (''Negoi'')
:<math>\gamma \sim H_n - \frac{\ln n + \ln(n+1)}{2} - \frac1{6n(n+1)} + \frac1{30n^2(n+1)^2} - \cdots</math> (''[[Ernesto Cesàro|Cesàro]]'')
The third formula is also called the [[Srinivasa Ramanujan|Ramanujan]] expansion.

===Exponential===
The constant {{math|''e{{isup|γ}}''}} is important in number theory. Some authors denote this quantity simply as {{math|''γ′''}}. {{math|''e{{isup|γ}}''}} equals the following [[limit of a sequence|limit]], where {{math|''p''<sub>''n''</sub>}} is the {{math|''n''}}th [[prime number]]:

:<math>e^\gamma = \lim_{n\to\infty}\frac1{\ln p_n} \prod_{i=1}^n \frac{p_i}{p_i-1}.</math>
This restates the third of [[Mertens' theorems]] {{harv|Weisstein|n.d.}}. The numerical value of {{math|''e{{isup|γ}}''}} is:
:{{gaps|1.78107|24179|90197|98523|65041|03107|17954|91696|45214|30343|...}} {{OEIS2C|id=A073004}}.

Other [[infinite product]]s relating to {{math|''e{{isup|γ}}''}} include:
:<math>\begin{align}
\frac{e^{1+\frac{\gamma}{2}}}{\sqrt{2\pi}} &= \prod_{n=1}^\infty e^{-1+\frac1{2n}}\left(1+\frac1{n}\right)^n \\
\frac{e^{3+2\gamma}}{2\pi} &= \prod_{n=1}^\infty e^{-2+\frac2{n}}\left(1+\frac2{n}\right)^n. \end{align}</math>
These products result from the [[Barnes G-function|Barnes {{math|''G''}}-function]].

In addition,
:<math>e^{\gamma} = \sqrt{\frac2{1}} \cdot \sqrt[3]{\frac{2^2}{1\cdot 3}} \cdot \sqrt[4]{\frac{2^3\cdot 4}{1\cdot 3^3}} \cdot \sqrt[5]{\frac{2^4\cdot 4^4}{1\cdot 3^6\cdot 5}} \cdots</math>
where the {{math|''n''}}th factor is the {{math|(''n'' + 1)}}th root of
:<math>\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}.</math>
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow {{harv|Sondow|2003}} using [[hypergeometric function]]s.

===Continued fraction===
The [[continued fraction]] expansion of {{mvar|γ}} is of the form {{nowrap|[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]}} {{OEIS2C|id=A002852}}, which has no ''apparent'' pattern. The continued fraction is known to have at least 470,000 terms {{harv|Havil|2003|p=97}}, and it has infinitely many terms [[if and only if]] {{mvar|γ}} is irrational.

==Generalizations==
[[File:Generalisation of Euler–Mascheroni constant.jpg|thumb|512px|right|{{math|abm(''x'') {{=}} ''γ''<sub>−''x''</sub>}}]]
''Euler's generalized constants'' are given by
:<math>\gamma_\alpha = \lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k^\alpha} - \int_1^n \frac1{x^\alpha}\,dx\right),</math>
for {{math|0 < ''α'' < 1}}, with {{mvar|γ}} as the special case {{math|''α'' {{=}} 1}} {{harv|Havil|2003|pp=117–118}}. This can be further generalized to
:<math>c_f = \lim_{n\to\infty}\left(\sum_{k=1}^n f(k) - \int_1^n f(x)\,dx\right)</math>
for some arbitrary decreasing function {{math|''f''}}. For example,
:<math>f_n(x) = \frac{(\ln x)^n}{x}</math>
gives rise to the [[Stieltjes constants]], and
:<math>f_a(x) = x^{-a}</math>
gives
:<math>\gamma_{f_a} = \frac{(a-1)\zeta(a)-1}{a-1}</math>
where again the limit
:<math>\gamma = \lim_{a\to 1}\left(\zeta(a) - \frac1{a-1}\right)</math>
appears.

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 {{harv|Ram Murty|Saradha|2010}}:
:<math>\gamma(a,q) = \lim_{x\to \infty}\left (\sum_{0<n\le x \atop n\equiv a \pmod q} \frac1{n}-\frac{\ln x}{q}\right).</math>

The basic properties are
:<math>\begin{align}
\gamma(0,q) &= \frac{\gamma -\ln q}{q}, \\
\sum_{a=0}^{q-1} \gamma(a,q)&=\gamma, \\
q\gamma(a,q) &= \gamma-\sum_{j=1}^{q-1}e^{-\frac{2\pi aij}{q}}\ln\left(1-e^{\frac{2\pi ij}{q}}\right),
\end{align}</math>
and if {{math|[[greatest common divisor|gcd]](''a'',''q'') {{=}} ''d''}} then
:<math>q\gamma(a,q) = \frac{q}{d}\gamma\left(\frac{a}{d},\frac{q}{d}\right)-\ln d.</math>

==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 and 31st-32nd decimal places; starting from the 20th digit, he calculated ...'''181'''12090082'''39''' when the correct value is ...'''065'''12090082'''40'''.

{| class="wikitable" style="margin: 1em auto 1em auto"
|+ '''Published Decimal Expansions of {{mvar|γ}}'''
! Date || Decimal digits || Author || Sources
|-
| 1734 || style="text-align:right;"| 5 || [[Leonhard Euler]] ||
|-
| 1735 || style="text-align:right;"| 15 || Leonhard Euler ||
|-
| 1781 || style="text-align:right;"| 16 || Leonhard Euler ||
|-
| 1790 || style="text-align:right;"| 32 || [[Lorenzo Mascheroni]], with 20-22 and 31-32 wrong ||
|-
| 1809 || style="text-align:right;"| 22 || [[Johann Georg von Soldner|Johann G. von Soldner]] ||
|-
| 1811 || style="text-align:right;"| 22 || [[Carl Friedrich Gauss]] ||
|-
| 1812 || style="text-align:right;"| 40 || [[Friedrich Bernhard Gottfried Nicolai]] ||
|-
| 1857 || style="text-align:right;"| 34 || Christian Fredrik Lindman ||
|-
| 1861 || style="text-align:right;"| 41 || Ludwig Oettinger ||
|-
| 1867 || style="text-align:right;"| 49 || [[William Shanks]] ||
|-
| 1871 || style="text-align:right;"| 99 || [[James Whitbread Lee Glaisher|James W.L. Glaisher]] ||
|-
| 1871 || style="text-align:right;"| 101 || William Shanks ||
|-
| 1877 || style="text-align:right;"| 262 || [[John Couch Adams|J. C. Adams]] ||
|-
| 1952 || style="text-align:right;"| 328 || [[John Wrench|John William Wrench Jr.]] ||
|-
| 1961 || style="text-align:right;"| {{val|1050|fmt=gaps}} || Helmut Fischer and [[Karl Longin Zeller|Karl Zeller]] ||
|-
| 1962 || style="text-align:right;"| {{val|1271|fmt=gaps}} || [[Donald Knuth]] ||
|-
| 1962 || style="text-align:right;"| {{val|3566|fmt=gaps}} || Dura W. Sweeney ||
|-
| 1973 || style="text-align:right;"| {{val|4879|fmt=gaps}} || William A. Beyer and [[Michael S. Waterman]] ||
|-
| 1977 || style="text-align:right;"| {{val|20700}} || [[Richard Brent (scientist)|Richard P. Brent]] ||
|-
| 1980 || style="text-align:right;"| {{val|30100}} || Richard P. Brent & [[Edwin McMillan|Edwin M. McMillan]] ||
|-
| 1993 || style="text-align:right;"| {{val|172000}} || [[Jonathan Borwein]] ||
|-
| 1999 || style="text-align:right;"| {{val|108000000}} || Patrick Demichel and Xavier Gourdon ||
|-
| 2009 || style="text-align:right;"| {{val|29844489545}} || Alexander J. Yee & Raymond Chan || {{harvnb|Yee|2011}}, {{harvnb|y-cruncher|2017}}
|-
| 2013 || style="text-align:right;"| {{val|119,377,958,182}} || Alexander J. Yee ||{{harvnb|Yee|2011}}, {{harvnb|y-cruncher|2017}}
|-
| 2016 || style="text-align:right;"| {{val|160,000,000,000}} || Peter Trueb || {{harvnb|y-cruncher|2017}}
|-
| 2016 || style="text-align:right;"| {{val|250,000,000,000}} || Ron Watkins || {{harvnb|y-cruncher|2017}}
|-
| 2017 || style="text-align:right;"| {{val|477,511,832,674}} || Ron Watkins ||{{harvnb|y-cruncher|2017}}
|}

==References==
* {{citation |last= Alabdulmohsin |first= Ibrahim M. |title = Summability Calculus. A Comprehensive Theory of Fractional Finite Sums |publisher = [[Springer-Verlag]] |isbn = 9783319746487 |year= 2018}}
* {{citation |last1= Blagouchine|first1= Iaroslav V.|title= Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results|journal= [[The Ramanujan Journal]]|date= 2014|volume= 35|issue= 1|pages= 21–110|ref= harv|doi=10.1007/s11139-013-9528-5|url=https://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten's_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results#.pdf}}
* {{citation|last1= Blagouchine|first1= Iaroslav V.|title= Expansions of generalized Euler's constants into the series of polynomials in {{math|π<sup>−2</sup>}} and into the formal enveloping series with rational coefficients only|journal= [[Journal of Number Theory|J. Number Theory]] |date= 2016|volume= 158|issue= |pages=365–396|arxiv=1501.00740|ref= harv|doi=10.1016/j.jnt.2015.06.012}}
* {{citation |last1= Blagouchine|first1= Iaroslav V.|title= Three notes on Ser's and Hasse's representations for the zeta-functions |journal= INTEGERS: The Electronic Journal of Combinatorial Number Theory |date= 2018 |volume= 18A|issue= #A3|pages= 1–45|url= http://math.colgate.edu/~integers/vol18a.html|bibcode= 2016arXiv160602044B|arxiv=1606.02044}}
*{{cite journal | last = Bretschneider | first = Carl Anton | year = 1837 | title = Theoriae logarithmi integralis lineamenta nova | url =https://zenodo.org/record/1448830/files/article.pdf | journal = Crelle's Journal | volume = 17 | issue = | pages = 257–285 |ref=harv|orig-year=submitted 1835|language=la}}
*{{cite book|last1=Caves |first1=Carlton M. |author-link1=Carlton M. Caves |last2=Fuchs |first2=Christopher A. |chapter=Quantum information: How much information in a state vector? |title=The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later |year=1996 |isbn=9780750303941 |oclc=36922834 |publisher=Israel Physical Society |arxiv=quant-ph/9601025|bibcode=1996quant.ph..1025C |ref=harv}}
*{{cite book|last=De Morgan|first=Augustus|authorlink=Augustus De Morgan|title=The differential and integral calculus|publisher=Baldwin and Craddoc|location=London|year=1836–1842|ref=harv}}
*{{cite journal|last=DeTemple|first=Duane W.|date=May 1993|title=A Quicker Convergence to Euler's Constant|journal=The American Mathematical Monthly|volume=100|issue=5|pages=468–470|doi=10.2307/2324300|issn=0002-9890|jstor=2324300|ref=harv}}
*{{cite journal|last=Glaisher|first=James Whitbread Lee|authorlink=James Whitbread Lee Glaisher|year=1910|title=On Dr. Vacca's series for {{mvar|γ}} | journal= Q. J. Pure Appl. Math |volume =41 | pages=365–368|ref=harv}}
*{{cite book |first = Julian |last = Havil |year = 2003 |title = Gamma: Exploring Euler's Constant |publisher = Princeton University Press |isbn = 978-0-691-09983-5 |ref=harv}}
*{{ cite journal| first1=G. H. |last1= Hardy |title=Note on Dr. Vacca's series for {{mvar|γ}} | journal= Q. J. Pure Appl. Math |volume =43 | pages=215–216 |year=1912|ref=harv}}
*{{cite journal | last1 = Kluyver | first1 = J.C. | year = 1927 | title = On certain series of Mr. Hardy | journal = Q. J. Pure Appl. Math. | volume = 50 | pages = 185–192|ref=harv}}
*{{citation|last1=Krämer|first1= Stefan |year=2005|title=Die Eulersche Konstante {{mvar|γ}} und verwandte Zahlen|publisher= University of Göttingen|location=Germany}}
*{{cite journal| last = Lagarias  | first = Jeffrey C.  |date=October 2013 | title = Euler's constant: Euler's work and modern developments | journal = [[Bulletin of the American Mathematical Society]]  | volume = 50  | issue = 4  | page = 556| doi=10.1090/s0273-0979-2013-01423-x| arxiv = 1303.1856| bibcode = 1994BAMaS..30..205W  |ref=harv}}
*{{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| issue=12 |pages=2671–2681|ref=harv}}
*{{Cite OEIS|A002852|Continued fraction for Euler's constant}}
* {{cite news|first1=Jonathan|last1=Sondow|year=1998|url=http://home.earthlink.net/~jsondow/id8.html|title=An antisymmetric formula for Euler's constant|journal=Mathematics Magazine|volume=71|pages=219–220|ref=harv|access-date=2006-05-29|archive-url=https://web.archive.org/web/20110604123534/http://home.earthlink.net/~jsondow/id8.html|archive-date=2011-06-04|url-status=dead}}
*{{cite journal | last1 = Sondow | first1 = Jonathan | year = 2002 | title = A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant| arxiv = math.NT/0211075| journal = Mathematica Slovaca | volume = 59 | issue = | pages = 307–314 | bibcode = 2002math.....11075S |ref=harv}} with an Appendix by [https://web.archive.org/web/20130523085959/http://wain.mi.ras.ru/zlobin/ Sergey Zlobin]
*{{cite arXiv | first1=Jonathan | last1= Sondow | year=2003 | eprint=math.CA/0306008 | title= An infinite product for {{math|''e{{sup|γ}}''}} via hypergeometric formulas for Euler's constant, {{mvar|γ}}|ref=harv}}
* {{citation|first=Jonathan|last1= Sondow|year=2003a | arxiv=math.NT/0209070 |title= Criteria for irrationality of Euler's constant| journal=Proceedings of the American Mathematical Society|volume= 131 |pages=3335–3344}}
*{{citation | last1 = Sondow | first1 = Jonathan | year = 2005 | title = Double integrals for Euler's constant and {{math|ln {{sfrac|4|π}}}} and an analog of Hadjicostas's formula | arxiv = math.CA/0211148 | journal = [[American Mathematical Monthly]] | volume = 112 | issue = 1| pages = 61–65 | doi=10.2307/30037385| jstor = 30037385}}
* {{citation|first1=Jonathan |last1= Sondow|year=2005a| eprint=math.NT/0508042 |title=New Vacca-type rational series for Euler's constant and its 'alternating' analog {{math|ln {{sfrac|4|π}}}}}}
* {{cite journal| first1=Jonathan |last1=Sondow | first2=Wadim | last2=Zudilin | year=2006 |arxiv=math.NT/0304021 |title= Euler's constant, {{math|''q''}}-logarithms, and formulas of Ramanujan and Gosper | doi=10.1007/s11139-006-0075-1 | volume=12 |issue=2 | journal=The Ramanujan Journal | pages=225–244|ref=harv}}
*{{Cite web|url=http://mathworld.wolfram.com/MertensConstant.html|title=Mertens Constant|first=Eric W.|last=Weisstein|authorlink=Eric W. Weisstein|website=mathworld.wolfram.com|year=n.d.|ref=harv}}
* {{cite web|url=http://www.numberworld.org/nagisa_runs/computations.html|title=Nagisa - Large Computations|last=Yee|first=Alexander J.|date=March 7, 2011|website=www.numberworld.org|ref=harv}}
* {{cite web|author=<!--Staff writer-->|url=http://www.numberworld.org/y-cruncher/records.html|title=Records Set by y-cruncher|date=August 24, 2017|access-date=April 30, 2018|ref=harv|website=www.numberworld.org}}

==Further reading==
*{{cite journal|author1=Borwein, Jonathan M. |author2=David M. Bradley |author3=Richard E. Crandall |title=Computational Strategies for the Riemann Zeta Function|journal=Journal of Computational and Applied Mathematics|year=2000|volume=121|issue=1–2 |pages=11|url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf|doi=10.1016/s0377-0427(00)00336-8|bibcode=2000JCoAM.121..247B|ref=harv}} Derives {{mvar|γ}} as sums over Riemann zeta functions.
*{{cite journal|first1=I. |last1=Gerst|title=Some series for Euler's constant
|year=1969|journal=Amer. Math. Monthly|doi=10.2307/2316370|volume=76|issue=3|pages=237–275|jstor=2316370|ref=harv}}
*{{cite journal|last=Glaisher|first=James Whitbread Lee|authorlink=James Whitbread Lee Glaisher|year=1872|title=On the history of Euler's constant|journal=Messenger of Mathematics|volume=1|pages=25–30|jfm=03.0130.01|ref=harv}}
*{{cite web|last=Gourdon|first=Xavier|last2=Seba|first2=P.|year=2002|url=http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html|title=Collection of formulas for Euler's constant, {{mvar|γ}}|ref=harv}}
*Gourdon, Xavier, and Sebah, P. (2004) "[http://numbers.computation.free.fr/Constants/Gamma/gamma.html The Euler constant: {{mvar|γ}}.]"
*{{ 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|ref=harv}}
*{{cite journal | last1 = Karatsuba | first1 = E.A. | year = 2000 | title = On the computation of the Euler constant {{mvar|γ|}} | url = | journal = Journal of Numerical Algorithms | volume = 24 | issue = 1–2| pages = 83–97 | doi = 10.1023/A:1019137125281 |ref=harv}}
*{{cite book|last=Knuth|first=Donald|authorlink=Donald Knuth|year=1997|title=[[The Art of Computer Programming]], Vol. 1|edition=3rd|publisher=Addison-Wesley|isbn=0-201-89683-4|ref=harv}}
*{{cite journal | last1 = Lerch | first1 = M. | year = 1897 | title = Expressions nouvelles de la constante d'Euler | url = | journal = Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften | volume = 42 | issue = | page = 5 |ref=harv}}
*{{citation|last=Mascheroni|first=Lorenzo|author-link=Lorenzo Mascheroni|year=1790|title=''"Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur"''|publisher=Galeati, Ticini|ref=harv}}
*{{cite journal| first1=D. H. | last1=Lehmer|year=1975|title=Euler constants for arithmetical progressions | journal=Acta Arith. |volume=27 |number=1| pages=125–142|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf| doi=10.4064/aa-27-1-125-142|ref=harv}}
*{{cite journal | last1 = Vacca | first1 = G. | year = 1926 | title = Nuova serie per la costante di Eulero, {{mvar|C}} = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche | url = | journal = Matematiche e Naturali | volume = 6 | issue = 3| pages = 19–20 |ref=harv}}

==External links==
*{{mathworld|urlname=Euler-MascheroniConstant|title=Euler–Mascheroni constant}}
*[http://home.earthlink.net/~jsondow/ Jonathan Sondow.]
*[http://www.ccas.ru/personal/karatsuba/algen.htm Fast Algorithms and the FEE Method], E.A. Karatsuba (2005)
*Further formulae which make use of the constant: [http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html Gourdon and Sebah (2004).]

{{DEFAULTSORT:Euler-Mascheroni Constant}}
[[Category:Mathematical constants]]
[[Category:Unsolved problems in mathematics]]
[[Category:Leonhard Euler]]