歐拉-馬斯刻若尼常數

歐拉-馬斯刻若尼常數是一個數學常數，定義為調和級數與自然對數的差值：

它的近似值為，
歐拉-馬斯刻若尼常數主要應用於數論。

**歷史**
該常數最先由瑞士數學家萊昂哈德·歐拉在1735年發表的文章De Progressionibus harmonicus observationes中定義。歐拉曾經使用作為它的符號，並計算出了它的前6位小數。1761年他又將該值計算到了16位小數。1790年，意大利數學家洛倫佐·馬斯凱羅尼引入了作為這個常數的符號，並將該常數計算到小數點後32位。但後來的計算顯示他在第20位的時候出現了錯誤。
目前尚不知道該常數是否為有理數，但是分析表明如果它是一個有理數，那麼它的分母位數將超過10242080。

**性質**

**與伽瑪函數的關係**
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**與ζ函數的關係**

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:。
:


:。


**積分**

:
:

。
 


**級數展開式**

.


的連分數展開式為：
 id=A002852.

**漸近展開式**




**已知位數**

**相關證明**
前面的放縮法主要證明了
 單調遞減並下有界限（0），所以極限存在。放縮法的結論需要使用ln(1+x)和ln(1-x)的泰勒級數展開進行證明。

**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. date=20091212002936 "
#Gourdon, Xavier, and Sebah, P. (2004) " The Euler constant: γ. date=20090428001402 "
#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.
# title= An infinite product for eγ via hypergeometric formulas for Euler's constant, γ
#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."
# 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. 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 date=20121024193226 
#
#year=1991
#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)
#  eprint=1303.1856, Bulletin of the American Mathematical Society 50 (4): 527-628 (2013)

**外部連結**
title=Euler-Mascheroni constant
*Krämer, Stefan " Euler's Constant γ=0.577... Its Mathematics and History."- Jonathan Sondow.- Fast Algorithms and the FEE Method date=20210402055902 , E.A. Karatsuba (2005)
*Further formulae which make use of the constant:  Gourdon and Sebah (2004). date=20091212002936