Here, 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 (Lagarias 2013). For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 (Bretschneider 1837, "γ = c = 0,577215 664901 532860 618112 090082 3.." on p. 260) and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842 (De Morgan 1836–1842, "γ" on p. 578)
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
- An upper bound on Shannon entropy in quantum information theory (Caves & Fuchs 1996)
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 (Havil 2003, p. 97). 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 functionEdit
This is equal to the limits:
Further limit results are (Krämer 2005):
Relation to the zeta functionEdit
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 can also be expressed as follows where A is the Glaisher–Kinkelin constant:
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 (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 (Sondow 2005a)
where N1(n) and N0(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.
for any . However, the rate of convergence of this expansion depends significantly on . In particular, exhibits much more rapid convergence than the conventional expansion (DeTemple 1993; Havil 2003, pp. 75-78). This is because
Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.
Euler showed that the following infinite series approaches γ:
In 1926 he found a second series:
Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series
where ψn(a) are the Bernoulli polynomials of the second kind, which are defined by the generating function
For any rational a this series contains rational terms only. For example, at a = 1, it becomes
A series related to the Akiyama-Tanigawa algorithm is
Series of prime numbers:
γ equals the following asymptotic formulas (where Hn is the nth harmonic number):
The third formula is also called the Ramanujan expansion.
Other infinite products relating to eγ include:
These products result from the Barnes G-function.
where the nth factor is the (n + 1)th root of
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, ...] OEIS: A002852, which has no apparent pattern. The continued fraction is known to have at least 470,000 terms (Havil 2003, p. 97), 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 (Havil 2003, pp. 117–118). 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.
Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class (Ram Murty & Saradha 2010):
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 20th–22nd and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.
|1790||32||Lorenzo Mascheroni, with 20-22 and 31-32 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||1050||Helmut Fischer and Karl Zeller|
|1962||3566||Dura W. Sweeney|
|1973||4879||William A. Beyer and Michael S. Waterman|
|1977||20700||Richard P. Brent|
|1980||30100||Richard P. Brent & Edwin M. McMillan|
|1999||108000000||Patrick Demichel and Xavier Gourdon|
|2009||29844489545||Alexander J. Yee & Raymond Chan||Yee 2011, y-cruncher 2017|
|2013||119377958182||Alexander J. Yee||Yee 2011, y-cruncher 2017|
|2016||160000000000||Peter Trueb||y-cruncher 2017|
|2016||250000000000||Ron Watkins||y-cruncher 2017|
|2017||477511832674||Ron Watkins||y-cruncher 2017|
- Alabdulmohsin, Ibrahim M. (2018), Summability Calculus. A Comprehensive Theory of Fractional Finite Sums, Springer-Verlag, ISBN 9783319746487
- Blagouchine, Iaroslav V. (2014), "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" (PDF), The Ramanujan Journal, 35 (1): 21–110, doi:10.1007/s11139-013-9528-5
- 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, 158: 365–396, arXiv:1501.00740, doi:10.1016/j.jnt.2015.06.012
- Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions", Integers (Electronic Journal of Combinatorial Number Theory), 18A (#A3): 1–45, arXiv:1606.02044, Bibcode:2016arXiv160602044B
- Bretschneider, Carl Anton (1837) [submitted 1835]. "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal (in Latin). 17: 257–285.
- Caves, Carlton M.; Fuchs, Christopher A. (1996). "Quantum information: How much information in a state vector?". The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later. Israel Physical Society. arXiv:quant-ph/9601025. Bibcode:1996quant.ph..1025C. ISBN 9780750303941. OCLC 36922834.
- De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc.
- DeTemple, Duane W. (May 1993). "A Quicker Convergence to Euler's Constant". The American Mathematical Monthly. 100 (5): 468–470. doi:10.2307/2324300. ISSN 0002-9890. JSTOR 2324300.
- Glaisher, James Whitbread Lee (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
- Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
- Hardy, G. H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
- Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
- Krämer, Stefan (2005), Die Eulersche Konstante γ und verwandte Zahlen, Germany: University of Göttingen
- Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. Bibcode:1994BAMaS..30..205W. doi:10.1090/s0273-0979-2013-01423-x.
- Ram Murty, M.; Saradha, N. (2010). "Euler–Lehmer constants and a conjecture of Erdos". JNT. 130 (12): 2671–2681. doi:10.1016/j.jnt.2010.07.004.
- Sloane, N. J. A. (ed.). "Sequence A002852 (Continued fraction for Euler's constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Sondow, Jonathan (1998). "An antisymmetric formula for Euler's constant". Mathematics Magazine. 71. pp. 219–220. Archived from the original on 2011-06-04. Retrieved 2006-05-29.
- Sondow, Jonathan (2002). "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant". Mathematica Slovaca. 59: 307–314. arXiv:math.NT/0211075. Bibcode:2002math.....11075S. with an Appendix by Sergey Zlobin
- Sondow, Jonathan (2003). "An infinite product for eγ via hypergeometric formulas for Euler's constant, γ". arXiv:math.CA/0306008.
- Sondow, Jonathan (2003a), "Criteria for irrationality of Euler's constant", Proceedings of the American Mathematical Society, 131: 3335–3344, arXiv:math.NT/0209070
- Sondow, Jonathan (2005), "Double integrals for Euler's constant and ln 4/ and an analog of Hadjicostas's formula", American Mathematical Monthly, 112 (1): 61–65, arXiv:math.CA/0211148, doi:10.2307/30037385, JSTOR 30037385
- Sondow, Jonathan (2005a), New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/, arXiv:math.NT/0508042
- Sondow, Jonathan; Zudilin, Wadim (2006). "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper". The Ramanujan Journal. 12 (2): 225–244. arXiv:math.NT/0304021. doi:10.1007/s11139-006-0075-1.
- Weisstein, Eric W. (n.d.). "Mertens Constant". mathworld.wolfram.com.
- Yee, Alexander J. (March 7, 2011). "Nagisa - Large Computations". www.numberworld.org.
- "Records Set by y-cruncher". www.numberworld.org. August 24, 2017. Retrieved April 30, 2018.
- 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 (1–2): 11. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8. Derives γ as sums over Riemann zeta functions.
- Gerst, I. (1969). "Some series for Euler's constant". Amer. Math. Monthly. 76 (3): 237–275. doi:10.2307/2316370. JSTOR 2316370.
- Glaisher, James Whitbread Lee (1872). "On the history of Euler's constant". Messenger of Mathematics. 1: 25–30. JFM 03.0130.01.
- Gourdon, Xavier; Seba, P. (2002). "Collection of formulas for Euler's constant, γ".
- Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ."
- 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". Journal of Numerical Algorithms. 24 (1–2): 83–97. doi:10.1023/A:1019137125281.
- Knuth, Donald (1997). The Art of Computer Programming, Vol. 1 (3rd ed.). Addison-Wesley. ISBN 0-201-89683-4.
- Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
- Mascheroni, Lorenzo (1790), "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur", Galeati, Ticini
- Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142.
- 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.