查看“歐拉-馬斯刻若尼常數”的源代码

{{NoteTA|G1=Math}}
'''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：

:<math>\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</math>

它的近似值为<math>\gamma \approx  0.57721 56649 01532 86060 65120 90082 40243 10421 59335</math><ref>{{OEIS2C|id=A001620}} oeis.org [2014-7-17]</ref>，

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<math>C</math>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{{le|洛倫佐·馬斯刻若尼|Lorenzo Mascheroni}}引入了<math>\gamma</math>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10<sup>242080</sup>。<ref name=Havil>Havil 2003 p 97.</ref>

== 性质 ==

=== 与伽玛函数的关系 ===

:<math> \ -\gamma = \Gamma'(1) = \Psi(1) </math>。

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

:<math> \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 ]</math>。

=== 与ζ函数的关系 ===

:<math>\gamma = \sum_{m=2}^{\infty} \frac{(-1)^m\zeta(m)}{m} </math>

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

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

:<math> \gamma = \frac{3}{2} - \ln 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m} [\zeta(m) - 1] </math>

::<math> = \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 ]</math>。

::<math> = \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 ] </math>

:<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 ) </math>

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

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

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

=== 积分 ===

:<math>\gamma = - \int_0^\infty { e^{-x} \ln x }\,dx = \int_\infty^ 0 { e^{-x} \ln x }\,dx </math><ref group="證明"><math>\gamma = - \int_0^\infty { e^{-x} \ln x }\,dx </math>的证明：<br>

首先根据放缩法（<math>\int_k^{k+1} \frac 1x \,dx < \frac 1k < \int_{k-1}^k \frac 1x \,dx</math>）容易知道，<math> \int_k^{k-1} \frac 1x \,dx - \frac 1k < \frac 1{k(k-1)}</math>，以及<math>\ln n < \sum_{k=1}^n \frac 1k < 1 + \ln n</math>。因此<math>\gamma</math>存在并有限。<br>

<math>\sum_{k=1}^n \frac{1}{k}</math><br>

<math>= \sum_{k=1}^n \int_0^1 t^{k-1} \,dt</math><br>

<math>= \int_0^1 \sum_{k=1}^n t^{k-1} \,dt</math><br>

<math>= \int_0^1 \frac {1 - t^n}{1 - t} \,dt</math><br>

<math>= \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)</math><br>

<math>= \int_n^0 \frac {1 - \left(1-\frac{x}{n}\right)^n}{\frac{x}{n}} \left(- \frac 1n\right) dx</math><br>

<math>= \int_0^n \frac {1 - \left(1-\frac{x}{n}\right)^n}{x} dx</math><br>

而<math>\ln n = \int_1^n \frac 1x \,dx,</math><br>

所以<math>\gamma = \lim_{n \to \infty} \left(\sum_{k=1}^n \frac{1}{k} - \ln n\right)</math><br>             

<math>= \lim_{n \to \infty} \left[ \int_0^n \frac {1 - (1-x/n)^n}{x} \,dx - \int_1^n \frac 1x \,dx \right]</math><br>

        <math>= \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]</math><br>

<math>= \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}</math>
（单调收敛定理）<br>

<math>= \int_0^1 \frac {1 - e^{-x}}{x} \,dx - \int_1^{\infty} \frac {e^{-x}}{x}</math><br>

<math>= \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}</math><br>

<math>= - \int_0^{\infty} e^{-x} \ln x \,dx.</math>
</ref><math> = - \int_0^1 { \ln\ln \frac{1}{x} }\,dx </math>

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

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

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

:<math> \int_0^\infty { e^{-x} \ln^2 x }\,dx  = \gamma^2 + \frac{\pi^2}{6}</math>。

: <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 )
 </math>

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

=== 级数展开式 ===

:<math>\gamma = \sum_{k=1}^\infty \left[ \frac{1}{k} - \ln \left( 1 + \frac{1}{k} \right) \right]</math>

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

:<math> \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</math>

<math> \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</math>

<math> \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</math>

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

<math> \gamma </math>的[[连分数]]展开式为：

:<math> \gamma = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]\, </math> {{OEIS|id=A002852}}.

=== 渐近展开式 ===

:<math>\gamma  \approx H_n  - \ln \left( n \right) - \frac{1}{{2n}} + \frac{1}{{12n^2 }} - \frac{1}{{120n^4 }} + ...</math>

:<math>\gamma  \approx H_n  - \ln \left( {n + \frac{1}{2} + \frac{1}{{24n}} - \frac{1}{{48n^3 }} + ...} \right)</math>

:<math>\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 }} - ...</math>

== 已知位数 ==
{| class="wikitable" style="margin: 1em auto 1em auto"
|+<math>\boldsymbol{\gamma}</math>'''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 || [[Lorenzo Mascheroni]]
|-
| 1809年 || 24 || [[Johann Georg von Soldner|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 Brent (scientist)|Richard P. Brent]]
|-
| 1980年 || 30,100 || [[Richard Brent (scientist)|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
|}

== 相关证明 ==
<references group="證明"/>

== 參考文獻 ==
<div class="reflist columns references-column-count references-column-count-2" style="{{column-count|2}} list-style-type: decimal;">
<references group=""></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
|volume=121
|pages=11
|url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf
|doi=10.1016/s0377-0427(00)00336-8}} 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.
#[[Jonathan Sondow|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.
#[[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 [https://web.archive.org/web/20130523085959/http://wain.mi.ras.ru/zlobin/ Sergey Zlobin], Mathematica Slovaca 59'': 307-314.
#{{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.
#[[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.&nbsp;25-30, JFM 03.0130.01
#Carl Anton Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle Journal, vol.17, p.&nbsp;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
 |year = 2003
 |title = Gamma: Exploring Euler's Constant
 |publisher = Princeton University Press
 |isbn = 0-691-09983-9
 }}
#{{ 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.&nbsp;83–97 (2000)
#M. Lerch, Expressions nouvelles de la constante d'Euler. Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften 42, 5 p. (1897)
# {{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)
</div>
== 外部連結 ==
*{{mathworld|urlname=Euler-MascheroniConstant|title=Euler-Mascheroni constant}}
*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.]"
*[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).]

[[Category:數學常數]]