欧拉-马斯刻若尼常数

欧拉-马斯刻若尼常数是一个数学常数，定义为调和级数与自然对数的差值：

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

提示：此条目的主题不是尤拉数。

欧拉-马斯刻若尼常数

欧拉-马斯刻若尼常数
蓝色区域的面积收敛到欧拉常数
识别
符号	${\displaystyle \gamma }$
位数数列编号	A001620
性质
定义	${\displaystyle \gamma =\lim _{n\rightarrow \infty }\left[\left(\sum _{k=1}^{n}{\frac {1}{k}}\right)-\ln(n)\right]}$ ${\displaystyle \gamma =\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx}$
连分数	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]
表示方式
值	${\displaystyle \gamma \approx }$0.57721566490153...
无穷级数	${\displaystyle \gamma =\sum _{k=1}^{\infty }\left[{\frac {1}{k}}-\ln \left(1+{\frac {1}{k}}\right)\right]}$
二进制	0.100100111100010001100111…
十进制	0.577215664901532860606512…
十六进制	0.93C467E37DB0C7A4D1BE3F81…
查 论 编

它的近似值为${\displaystyle \gamma \approx 0.577215664901532860606512090082402431042159335}$^[1]，

欧拉-马斯刻若尼常数主要应用于数论。

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${\displaystyle C}$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，意大利数学家洛伦佐·马斯凯罗尼引入了${\displaystyle \gamma }$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。^[2]

性质

与伽玛函数的关系

${\displaystyle \ -\gamma =\Gamma '(1)=\Psi (1)}$。

${\displaystyle \gamma =\lim _{x\to \infty }\left[x-\Gamma \left({\frac {1}{x}}\right)\right]}$。

${\displaystyle \gamma =\lim _{n\to \infty }\left[{\frac {\Gamma ({\frac {1}{n}})\Gamma (n+1)\,n^{1+{\frac {1}{n}}}}{\Gamma (2+n+{\frac {1}{n}})}}-{\frac {n^{2}}{n+1}}\right]}$。

与ζ函数的关系

${\displaystyle \gamma =\sum _{m=2}^{\infty }{\frac {(-1)^{m}\zeta (m)}{m}}}$

${\displaystyle =\ln \left({\frac {4}{\pi }}\right)+\sum _{m=1}^{\infty }{\frac {(-1)^{m-1}\zeta (m+1)}{2^{m}(m+1)}}}$。

${\displaystyle \lim _{\varepsilon \to 0}{\frac {\zeta (1+\varepsilon )+\zeta (1-\varepsilon )}{2}}=\gamma }$

${\displaystyle \gamma ={\frac {3}{2}}-\ln 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}[\zeta (m)-1]}$

${\displaystyle =\lim _{n\to \infty }\left[{\frac {2\,n-1}{2\,n}}-\ln \,n+\sum _{k=2}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right]}$。

${\displaystyle =\lim _{n\to \infty }\left[{\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{m\,n}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\,\ln 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right]}$

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

${\displaystyle \gamma =\lim _{x\to \infty }\left[x-\Gamma \left({\frac {1}{x}}\right)\right]}$

${\displaystyle =\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right)}$。

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

积分

${\displaystyle \gamma =-\int _{0}^{\infty }{e^{-x}\ln x}\,dx=\int _{\infty }^{0}{e^{-x}\ln x}\,dx}$^{[证明 1]}${\displaystyle =-\int _{0}^{1}{\ln \ln {\frac {1}{x}}}\,dx}$

${\displaystyle =\int _{0}^{\infty }{\left({\frac {1}{1-e^{-x}}}-{\frac {1}{x}}\right)e^{-x}}\,dx}$

${\displaystyle =\int _{0}^{\infty }{{\frac {1}{x}}\left({\frac {1}{1+x}}-e^{-x}\right)}\,dx}$

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

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

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

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

级数展开式

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

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

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

${\displaystyle \gamma +\zeta (2)=\sum _{k=1}^{\infty }{\frac {1}{k\lfloor {\sqrt {k}}\rfloor ^{2}}}=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}\left({\tfrac {1}{4}}+\dots +{\tfrac {1}{8}}\right)+{\tfrac {1}{9}}\left({\tfrac {1}{9}}+\dots +{\tfrac {1}{15}}\right)+\dots }$

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

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

${\displaystyle \gamma }$的连分数展开式为：

${\displaystyle \gamma =[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,...]\,}$ （OEIS数列A002852）.

渐近展开式

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

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

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

已知位数

${\displaystyle {\boldsymbol {\gamma }}}$的已知位数

日期	位数	计算者
1734年	5	莱昂哈德·欧拉
1736年	15	莱昂哈德·欧拉
1790年	19	洛伦佐·马斯凯罗尼
1809年	24	Johann G. von Soldner
1812年	40	F.B.G. Nicolai
1861年	41	Oettinger
1869年	59	William Shanks
1871年	110	William Shanks
1878年	263	约翰·柯西·亚当斯
1962年	1,271	高德纳
1962年	3,566	D.W. Sweeney
1977年	20,700	Richard P. Brent
1980年	30,100	Richard P. Brent和埃德温·麦克米伦
1993年	172,000	Jonathan Borwein
1997年	1,000,000	Thomas Papanikolaou
1998年12月	7,286,255	Xavier Gourdon
1999年10月	108,000,000	Xavier Gourdon和Patrick Demichel
2006年7月16日	2,000,000,000	Shigeru Kondo和Steve Pagliarulo
2006年12月8日	116,580,041	Alexander J. Yee
2007年7月15日	5,000,000,000	Shigeru Kondo和Steve Pagliarulo
2008年1月1日	1,001,262,777	Richard B. Kreckel
2008年1月3日	131,151,000	Nicholas D. Farrer
2008年6月30日	10,000,000,000	Shigeru Kondo和Steve Pagliarulo
2009年1月18日	14,922,244,771	Alexander J. Yee和Raymond Chan
2009年3月13日	29,844,489,545	Alexander J. Yee和Raymond Chan
2013年	119,377,958,182	Alexander J. Yee
2016年	160,000,000,000	Peter Trueb
2016年	250,000,000,000	Ron Watkins
2017年	477,511,832,674	Ron Watkins
2020年	600,000,000,100	Seungmin Kim和Ian Cutress

相关证明

1. ${\displaystyle \gamma =-\int _{0}^{\infty }{e^{-x}\ln x}\,dx}$的证明：
   首先根据放缩法（${\displaystyle \int _{k}^{k+1}{\frac {1}{x}}\,dx<{\frac {1}{k}}<\int _{k-1}^{k}{\frac {1}{x}}\,dx}$）容易知道，${\displaystyle \int _{k}^{k-1}{\frac {1}{x}}\,dx-{\frac {1}{k}}<{\frac {1}{k(k-1)}}}$，以及${\displaystyle \ln n<\sum _{k=1}^{n}{\frac {1}{k}}<1+\ln n}$。因此${\displaystyle \gamma }$存在并有限。
   ${\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}}$
   ${\displaystyle =\sum _{k=1}^{n}\int _{0}^{1}t^{k-1}\,dt}$
   ${\displaystyle =\int _{0}^{1}\sum _{k=1}^{n}t^{k-1}\,dt}$
   ${\displaystyle =\int _{0}^{1}{\frac {1-t^{n}}{1-t}}\,dt}$
   ${\displaystyle =\int _{n}^{0}{\frac {1-\left(1-{\frac {x}{n}}\right)^{n}}{1-\left(1-{\frac {x}{n}}\right)}}d\left(1-{\tfrac {x}{n}}\right)}$
   ${\displaystyle =\int _{n}^{0}{\frac {1-\left(1-{\frac {x}{n}}\right)^{n}}{\frac {x}{n}}}\left(-{\frac {1}{n}}\right)dx}$
   ${\displaystyle =\int _{0}^{n}{\frac {1-\left(1-{\frac {x}{n}}\right)^{n}}{x}}dx}$
   而${\displaystyle \ln n=\int _{1}^{n}{\frac {1}{x}}\,dx,}$
   所以${\displaystyle \gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n\right)}$
   ${\displaystyle =\lim _{n\to \infty }\left[\int _{0}^{n}{\frac {1-(1-x/n)^{n}}{x}}\,dx-\int _{1}^{n}{\frac {1}{x}}\,dx\right]}$
   ${\displaystyle =\lim _{n\to \infty }\left[\int _{0}^{1}{\frac {1-(1-x/n)^{n}}{x}}\,dx-\int _{1}^{n}{\frac {(1-x/n)^{n}}{x}}\right]}$
   ${\displaystyle =\int _{0}^{1}{\frac {1-\lim _{n\to \infty }(1-x/n)^{n}}{x}}\,dx-\int _{1}^{\infty }{\frac {\lim _{n\to \infty }(1-x/n)^{n}}{x}}}$ （单调收敛定理）
   ${\displaystyle =\int _{0}^{1}{\frac {1-e^{-x}}{x}}\,dx-\int _{1}^{\infty }{\frac {e^{-x}}{x}}}$
   ${\displaystyle =\left.(1-e^{-x})\ln x\right|_{0}^{1}-\int _{0}^{1}\ln x\,d(1-e^{-x})-\left.e^{-x}\ln x\right|_{1}^{\infty }+\int _{1}^{\infty }\ln x\,de^{-x}}$
   ${\displaystyle =-\int _{0}^{\infty }e^{-x}\ln x\,dx.}$

前面的放缩法主要是证明了

${\displaystyle \left[\left(\sum _{k=1}^{n}{\frac {1}{k}}\right)-\ln(n)\right]}$ 是单调递减并下有界限（0），所有极限存在。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。

参考文献

1. A001620 oeis.org [2014-7-17]
2. Havil 2003 p 97.

1. Borwein, Jonathan M., David M. Bradley, Richard E. Crandall. Computational Strategies for the Riemann Zeta Function (PDF). Journal of Computational and Applied Mathematics. 2000, 121: 11 [2014-07-17]. doi:10.1016/s0377-0427(00)00336-8. （原始内容 (PDF)存档于2006-09-25）. Derives γ as sums over Riemann zeta functions.
2. Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ. （页面存档备份，存于互联网档案馆）"
3. Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ. （页面存档备份，存于互联网档案馆）"
4. Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. ISBN 978-0-201-89683-1
5. Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Universität Göttingen.
6. Sondow, Jonathan (1998) "An antisymmetric formula for Euler's constant," Mathematics Magazine 71: 219-220.
7. Sondow, Jonathan (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.
8. Sondow, Jonathan. An infinite product for e^γ via hypergeometric formulas for Euler's constant, γ. 2003. arXiv:math.CA/0306008 .
9. Sondow, Jonathan (2003a) "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131: 3335-3344.
10. Sondow, Jonathan (2005) "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula," American Mathematical Monthly 112: 61-65.
11. Sondow, Jonathan (2005) "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π."
12. Sondow, Jonathan; Zudilin, Wadim. Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper. 2006. arXiv:math.NT/0304021 . Ramanujan Journal 12: 225-244.
13. 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.
14. 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
15. Carl Anton Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle Journal, vol.17, p. 257-285 (submitted 1835)
16. Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.
17. 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 （页面存档备份，存于互联网档案馆）
18. Havil, Julian. Gamma: Exploring Euler's Constant. Princeton University Press. 2003. ISBN 0-691-09983-9.
19. Karatsuba, E. A. Fast evaluation of transcendental functions. Probl. Inf. Transm. 1991, 27 (44): 339–360.
20. E.A. Karatsuba, On the computation of the Euler constant γ, J. of Numerical Algorithms Vol.24, No.1-2, pp. 83–97 (2000)
21. M. Lerch, Expressions nouvelles de la constante d'Euler. Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften 42, 5 p. (1897)
22. Lagarias, Jeffrey C. Euler's constant: Euler's work and modern developments. arXiv:1303.1856 ., Bulletin of the American Mathematical Society 50 (4): 527-628 (2013)

外部链接

• 埃里克·韦斯坦因. Euler-Mascheroni constant. MathWorld.
• Krämer, Stefan "Euler's Constant γ=0.577... Its Mathematics and History."
• Jonathan Sondow.
• Fast Algorithms and the FEE Method （页面存档备份，存于互联网档案馆）, E.A. Karatsuba (2005)
• Further formulae which make use of the constant: Gourdon and Sebah (2004). （页面存档备份，存于互联网档案馆）