Here, ⌊x⌋ represents the floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is
|Continued fraction||[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ]
(It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic.
Shown in linear notation)
The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.
The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):
- Expressions involving the exponential integral*
- The Laplace transform* of the natural logarithm
- The first term of the Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*
- Calculations of the digamma function
- A product formula for the gamma function
- An inequality for Euler's totient function
- The growth rate of the divisor function
- In Dimensional regularization of Feynman diagrams in Quantum Field Theory
- The calculation of the Meissel–Mertens constant
- The third of Mertens' theorems*
- Solution of the second kind to Bessel's equation
- In the regularization/renormalization of the Harmonic series as a finite value
- The mean of the Gumbel distribution
- The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
- The answer to the coupon collector's problem*
- In some formulations of Zipf's law
- A definition of the cosine integral*
- Lower bounds to a prime gap.
The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Continued fraction analysis reveals that if γ is rational, its denominator must be greater than 10242080. The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics. Also see Sondow (2003a).
Relation to gamma function
This is equal to the limits:
Further limit results are (Krämer, 2005):
Relation to the zeta function
Other series related to the zeta function include:
The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is well-suited for efficient computation of the constant to high precision.
Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit (Sondow, 1998):
and de la Vallée-Poussin's formula
where are ceiling brackets.
Closely related to this is the rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:
where 0 < ε < 1/.
γ equals the value of a number of definite integrals:
where Hx is the fractional harmonic number.
Definite integrals in which γ appears include:
An interesting comparison by J. Sondow (2005) is the double integral and alternating series
It shows that ln 4/ may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series (see Sondow 2005 #2)
where N1(n) and N0(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.
We have also Catalan's 1875 integral (see Sondow and Zudilin)
Euler showed that the following infinite series approaches γ:
The series for γ is equivalent to a series Nielsen found in 1897:
In 1910, Vacca found the closely related series:
In 1926 he found a second series:
where Gn are Gregory coefficients.
Another important expansion with the Gregory coefficients involving Euler's constant is:
and is convergent for all n.
Series of prime numbers:
Series relating to square roots:
γ equals the following asymptotic formulas (where Hn is the nth harmonic number):
The third formula is also called the Ramanujan expansion.
- 1.78107241799019798523650410310717954916964521430343… A073004.
Other infinite products relating to eγ include:
These products result from the Barnes G-function.
We also have
where the nth factor is the (n + 1)th root of
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.
The continued fraction expansion of γ is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] A002852, of which there is no apparent pattern. The continued fraction is known to have at least 470,000 terms, and it has infinitely many terms if and only if γ is irrational.
Euler's generalized constants are given by
for 0 < α < 1, with γ as the special case α = 1. This can be further generalized to
for some arbitrary decreasing function f. For example,
gives rise to the Stieltjes constants, and
where again the limit
A two-dimensional limit generalization is the Masser–Gramain constant.
The basic properties are
and if gcd(a,q) = d then
Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 19th–21st and 32nd decimal places; starting from the 19th digit, he calculated …1811209008239 when the correct value is …0651209008240.
|1790||32||Lorenzo Mascheroni, with 19-21 and 32 wrong|
|1809||22||Johann G. von Soldner|
|1811||22||Carl Friedrich Gauss|
|1812||40||Friedrich Bernhard Gottfried Nicolai|
|1857||34||Christian Fredrik Lindman|
|1871||99||James W.L. Glaisher|
|1877||262||J. C. Adams|
|1952||328||John William Wrench Jr.|
|1961||0501||Helmut Fischer and Karl Zeller|
|1962||5663||Dura W. Sweeney|
|1973||8794||William A. Beyer and Michael S. Waterman|
|1977||70020||Richard P. Brent|
|1980||10030||Richard P. Brent & Edwin M. McMillan|
|1999||000000108||Patrick Demichel and Xavier Gourdon|
|2009||84448954529||Alexander J. Yee & Raymond Chan|
|2013||377958182119||Alexander J. Yee|
- Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments" (PDF). Bulletin of the American Mathematical Society. 50 (4): 556. doi:10.1090/s0273-0979-2013-01423-x.
- Carl Anton Bretschneider: Theoriae logarithmi integralis lineamenta nova (13 October 1835), Journal für die reine und angewandte Mathematik 17, 1837, pp. 257–285 (in Latin; "γ = c = 0,577215 664901 532860 618112 090082 3.." on [Euler–Mascheroni constant p. 260])
- Augustus De Morgan: The differential and integral calculus, Baldwin and Craddock, London 1836–1842 ("γ" on p. 578)
- Havil 2003 p. 97.
- http://mathworld.wolfram.com/MertensConstant.html (14)
- Havil, pp. 117–118
- Ram Murty, M.; Saradha, N. (2010). "Euler–Lehmer constants and a conjecture of Erdos". JNT. 130: 2671–2681. doi:10.1016/j.jnt.2010.07.004.
- Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142.
- Nagisa – Large Computations
- Blagouchine, Iaroslav V. (2014), "Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results", The Ramanujan Journal, 35 (1): 21–110, doi:10.1007/s11139-013-9528-5 PDF
- Blagouchine, Iaroslav V. (2016), "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only", J. Number Theory (Elsevier), 158: 365–396, arXiv: , doi:10.1016/j.jnt.2015.06.012
- Borwein, Jonathan M.; David M. Bradley; Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function" (PDF). Journal of Computational and Applied Mathematics. 121: 11. doi:10.1016/s0377-0427(00)00336-8. Derives γ as sums over Riemann zeta functions.
- Carl Anton, Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle Journal. 17: 257–285. (submitted 1835)
- Gerst, I. (1969). "Some series for Euler's constant". Am. Math. Monthly. 76: 237–275. doi:10.2307/2316370.
- 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 0-201-89683-4
- Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Ph.D. Thesis, University of Göttingen, Germany.
- Sondow, Jonathan (1998). "An antisymmetric formula for Euler's constant". Mathematics Magazine. 71. pp. 219–220.
- Sondow, Jonathan (2002). "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant". Mathematica Slovaca. 59: 307–314. arXiv: . with an Appendix by Sergey Zlobin
- Sondow, Jonathan (2003). "An infinite product for eγ via hypergeometric formulas for Euler's constant, γ". arXiv: .
- Sondow, Jonathan (2003). "Criteria for irrationality of Euler's constant". Proceedings of the American Mathematical Society. 131. pp. 3335–3344. arXiv: .
- Sondow, Jonathan (2005). "Double integrals for Euler's constant and ln 4/ and an analog of Hadjicostas's formula". American Mathematical Monthly. 112: 61–65. arXiv: . doi:10.2307/30037385.
- Sondow, Jonathan (2005). "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/". arXiv: .
- Sondow, Jonathan; Zudilin, Wadim (2006). "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper". arXiv: . Ramanujan Journal 12: 225-244.
- Vacca, G. (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. 1: 25–30., JFM 03.0130.01
- 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 Euler–Mascheroni constant at Google Books
- Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 0-691-09983-9.
- Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (44): 339–360.
- Karatsuba, E.A. (2000). "On the computation of the Euler constant γ". J. of Numerical Algorithms. 24 (1-2): 83–97.
- Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
- Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. doi:10.1090/s0273-0979-2013-01423-x.