{{Infobox number
| name=歐拉-馬斯刻若尼常數
| nav=no
| is integer = no
| image_name=gamma-area.svg
| image_caption=藍色區域的面積收斂到歐拉常數
| number=
| symbol=\gamma
| value=\gamma\approx0.57721566490153...
| OEIS=A001620
| 發現者=
| define=\gamma = \lim_{n \rightarrow \infty } \left[ \left(
\sum_{k=1}^n \frac{1}{k} \right) - \ln(n) \right]
\gamma = \int_1^\infty\left({1\over\lfloor x\rfloor} - {1\over x}\right)\,dx
| basedata =
| 連分數=[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]
| series=\gamma = \sum_{k=1}^\infty \left[ \frac{1}{k} - \ln \left( 1 + \frac{1}{k} \right) \right]
}}
歐拉-馬斯刻若尼常數是一个数学常数，定义为调和级数与自然对数的差值：

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

它的近似值为\gamma \approx 0.57721 56649 01532 86060 65120 90082 40243 10421 59335，

歐拉-馬斯刻若尼常數主要应用于数论。

历史
该常数最先由瑞士数学家莱昂哈德·欧拉在1735年发表的文章De Progressionibus harmonicus observationes中定义。欧拉曾经使用C作为它的符号，并计算出了它的前6位小数。1761年他又将该值计算到了16位小数。1790年，意大利数学家洛倫佐·馬斯凱羅尼引入了\gamma作为这个常数的符号，并将该常数计算到小数点后32位。但后来的计算显示他在第20位的时候出现了错误。

目前尚不知道该常数是否为有理数，但是分析表明如果它是一个有理数，那么它的分母位数将超过10242080。

性质
与伽玛函数的关系
: \ -\gamma = \Gamma'(1) = \Psi(1) 。

: \gamma = \lim_{x \to \infty} \left [ x - \Gamma \left ( \frac{1}{x} \right ) \right ]。

: \gamma = \lim_{n \to \infty} \left [ \frac{ \Gamma(\frac{1}{n}) \Gamma(n+1)\, n^{1+\frac{1}{n}}}{\Gamma(2+n+\frac{1}{n})} - \frac{n^2}{n+1} \right ]。

与ζ函数的关系
:\gamma = \sum_{m=2}^{\infty} \frac{(-1)^m\zeta(m)}{m}

:= \ln \left ( \frac{4}{\pi} \right ) + \sum_{m=1}^{\infty} \frac{(-1)^{m-1} \zeta(m+1)}{2^m (m+1)}。

: \lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2} = \gamma

: \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 ]

: \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 )

: \gamma = \lim_{x \to \infty} \left [ x - \Gamma \left ( \frac{1}{x} \right ) \right ]

:: = \lim_{n \to \infty} \frac{1}{n}\, \sum_{k=1}^n \left ( \left \lceil \frac{n}{k} \right \rceil - \frac{n}{k} \right )。

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

积分
:\gamma = - \int_0^\infty { e^{-x} \ln x }\,dx = \int_\infty^ 0 { e^{-x} \ln x }\,dx = - \int_0^1 { \ln\ln \frac{1}{x} }\,dx

:: = \int_0^\infty {\left (\frac{1}{1 - e^{-x}} - \frac{1}{x} \right )e^{ - x} }\,dx

:: = \int_0^\infty { \frac{1}{x} \left ( \frac{1}{1+x} - e^{ - x} \right ) }\,dx

: \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}。

: \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 )

: \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} = \gamma

级数展开式
:\gamma = \sum_{k=1}^\infty \left[ \frac{1}{k} - \ln \left( 1 + \frac{1}{k} \right) \right]

\gamma = 1 - \sum_{k=2}^{\infty}(-1)^k\frac{\lfloor\log_2 k\rfloor}{k+1} .

: \gamma = \sum_{k=2}^\infty (-1)^k \frac{ \left \lfloor \log_2 k \right \rfloor}{k}
= \tfrac12-\tfrac13

• 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right)
• 3\left(\tfrac18 - \dots - \tfrac1{15}\right) + \dots

\gamma + \zeta(2) = \sum_{k=1}^{\infty} \frac1{k\lfloor\sqrt{k}\rfloor^2}
= 1 + \tfrac12 + \tfrac13 + \tfrac14\left(\tfrac14 + \dots + \tfrac18\right)

• \tfrac19\left(\tfrac19 + \dots + \tfrac1{15}\right) + \dots

\gamma = \sum_{k=2}^{\infty} \frac{k - \lfloor\sqrt{k}\rfloor^2}{k^2\lfloor\sqrt{k}\rfloor^2}
= \tfrac1{2^2} + \tfrac2{3^2}

• \tfrac1{2^2}\left(\tfrac1{5^2} + \tfrac2{6^2} + \tfrac3{7^2} + \tfrac4{8^2}\right)
• \tfrac1{3^2}\left(\tfrac1{10^2} + \dots + \tfrac6{15^2}\right) + \dots

: \gamma = \int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx

\gamma 的连分数展开式为：

: \gamma = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]\, .

渐近展开式
:\gamma \approx H_n - \ln \left( n \right) - \frac{1} + \frac{1} - \frac{1} + ...

:\gamma \approx H_n - \ln \left( {n + \frac{1}{2} + \frac{1} - \frac{1} + ...} \right)

:\gamma \approx 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 }} - ...

已知位数
相关证明
前面的放缩法主要证明了
: \left[ \left(
\sum_{k=1}^n \frac{1}{k} \right) - \ln(n) \right] 单调递减并下有界限（0），所以极限存在。放缩法的结论需要使用ln(1+x)和ln(1-x)的泰勒级数展开进行证明。

參考文獻

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.] "
#Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. ISBN 978-0-201-89683-1
#Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Universität Göttingen.
#Sondow, Jonathan (1998) "[https://web.archive.org/web/20110604123534/http://home.earthlink.net/~jsondow/id8.html An antisymmetric formula for Euler's constant,]" Mathematics Magazine 71: 219-220.
#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 [https://web.archive.org/web/20130523085959/http://wain.mi.ras.ru/zlobin/ Sergey Zlobin], Mathematica Slovaca 59*: 307-314.
#
#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.
#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.
#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.]"

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
#
#
#E.A. Karatsuba, On the computation of the Euler constant γ, J. of Numerical Algorithms Vol.24, No.1-2, pp. 83–97 (2000)
#M. Lerch, Expressions nouvelles de la constante d'Euler. Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften 42, 5 p. (1897)

, Bulletin of the American Mathematical Society 50 (4): 527-628 (2013)

外部連結
*
*Krämer, Stefan "[https://web.archive.org/web/20131017040641/http://www.math.uni-goettingen.de/skraemer/gamma.html Euler's Constant γ=0.577... Its Mathematics and History.]"
*[https://web.archive.org/web/20071210200421/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).]