歐拉-馬斯刻若尼常數

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：

它的近似值为，

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 

。

。

。

与ζ函数的关系 

。

。

。

积分 

。

级数展开式 

.

的连分数展开式为：

 .

渐近展开式

已知位数

相关证明

參考文獻 
 Derives γ as sums over Riemann zeta functions.
Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ. "
Gourdon, Xavier, and Sebah, P. (2004) "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) "An antisymmetric formula for Euler's constant," Mathematics Magazine 71: 219-220.
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.

Sondow, Jonathan (2003a) "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131: 3335-3344.
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.
Sondow, Jonathan (2005) "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 "Euler's Constant γ=0.577... Its Mathematics and History."
Jonathan Sondow.
Fast Algorithms and the FEE Method , E.A. Karatsuba (2005)
Further formulae which make use of the constant: Gourdon and Sebah (2004). 

數學常數
莱昂哈德·欧拉