Euler's constant
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The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (γ).
It is defined as the limiting difference between the harmonic series and the natural logarithm:
Here, represents the floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:
Binary | 0.1001001111000100011001111110001101111101... |
Decimal | 0.5772156649015328606065120900824024310421... |
Hexadecimal | 0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3A... |
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) Source: Sloane |
History
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)
Appearances
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)
Properties
The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10244663.[1][2] 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
γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:
This is equal to the limits:
Further limit results are (Krämer 2005):
A limit related to the beta function (expressed in terms of gamma functions) is
Relation to the zeta function
γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:
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 ζ(s,k) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Hn. Expanding some of the terms in the Hurwitz zeta function gives:
where 0 < ε < 1/252n6.
γ can also be expressed as follows where A is the Glaisher–Kinkelin constant:
γ can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:
Integrals
γ equals the value of a number of definite integrals:
where Hx is the fractional harmonic number.
Definite integrals in which γ appears include:
One can express γ using a special case of Hadjicostas's formula as a double integral (Sondow 2003a) and (Sondow 2005) with equivalent series:
An interesting comparison by (Sondow 2005) is the double integral and alternating series
It shows that ln 4/π 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.
We have also Catalan's 1875 integral (see Sondow & Zudilin 2006)
Series expansions
In general,
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
while
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:
The series for γ is equivalent to a series Nielsen found in 1897 (Krämer 2005, Blagouchine 2016):
In 1910, Vacca found the closely related series (Vacca 1910 ,[citation not found] Glaisher 1910, Hardy 1912, Vacca 1925 ,[citation not found] Kluyver 1927, Krämer 2005, Blagouchine 2016)
where log2 is the logarithm to base 2 and ⌊ ⌋ is the floor function.
In 1926 he found a second series:
From the Malmsten–Kummer expansion for the logarithm of the gamma function (Blagouchine 2014) we get:
An important expansion for Euler's constant is due to Fontana and Mascheroni
where Gn are Gregory coefficients (Krämer 2005, Blagouchine 2016, Blagouchine 2018) This series is the special case of the expansions
convergent for
A similar series with the Cauchy numbers of the second kind Cn is (Blagouchine 2016; Alabdulmohsin 2018, pp. 147–148)
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
see OEIS: A302120 and OEIS: A302121. Other series with the same polynomials include these examples:
and
where Γ(a) is the gamma function (Blagouchine 2018).
A series related to the Akiyama-Tanigawa algorithm is
where Gn(2) are the Gregory coefficients of the second order (Blagouchine 2018).
Series of prime numbers:
Asymptotic expansions
γ equals the following asymptotic formulas (where Hn is the nth harmonic number):
- (Euler)
- (Negoi)
- (Cesàro)
The third formula is also called the Ramanujan expansion.
Alabdulmohsin 2018, pp. 147–148 derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1):
Exponential
The constant eγ is important in number theory. Some authors denote this quantity simply as γ′. eγ equals the following limit, where pn is the nth prime number:
This restates the third of Mertens' theorems (Weisstein n.d.). The numerical value of eγ is:
Other infinite products relating to eγ include:
These products result from the Barnes G-function.
In addition,
where the nth factor is the (n + 1)th root of
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (Sondow 2003) using hypergeometric functions.
Continued fraction
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 475,006 terms,[1] and it has infinitely many terms if and only if γ is irrational.
Generalizations
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Generalisation_of_Euler%E2%80%93Mascheroni_constant.jpg/512px-Generalisation_of_Euler%E2%80%93Mascheroni_constant.jpg)
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
gives
where again the limit
appears.
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
Published digits
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.
Date | Decimal digits | Author | Sources |
---|---|---|---|
1734 | 5 | Leonhard Euler | |
1735 | 15 | Leonhard Euler | |
1781 | 16 | Leonhard Euler | |
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 | |
1861 | 41 | Ludwig Oettinger | |
1867 | 49 | William Shanks | |
1871 | 99 | James W.L. Glaisher | |
1871 | 101 | William Shanks | |
1877 | 262 | J. C. Adams | |
1952 | 328 | John William Wrench Jr. | |
1961 | 1050 | Helmut Fischer and Karl Zeller | |
1962 | 1271 | Donald Knuth | |
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 | |
1993 | 172000 | Jonathan Borwein | |
1999 | 108000000 | Patrick Demichel and Xavier Gourdon | |
March 13, 2009 | 29844489545 | Alexander J. Yee & Raymond Chan | Yee 2011, y-cruncher 2017 |
December 22, 2013 | 119377958182 | Alexander J. Yee | Yee 2011, y-cruncher 2017 |
March 15, 2016 | 160000000000 | Peter Trueb | y-cruncher 2017 |
May 18, 2016 | 250000000000 | Ron Watkins | y-cruncher 2017 |
August 23, 2017 | 477511832674 | Ron Watkins | y-cruncher 2017 |
May 26, 2020 | 600000000100 | Seungmin Kim & Ian Cutress[3] | [4] |
Notes
- ^ a b Haible, Bruno; Papanikolaou, Thomas (1998). Buhler, Joe P. (ed.). "Fast multiprecision evaluation of series of rational numbers". Algorithmic Number Theory. Lecture Notes in Computer Science. Springer Berlin Heidelberg: 338–350. doi:10.1007/bfb0054873. ISBN 978-3-540-69113-6.
- ^ *Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis). Universität des Saarlandes.
- ^ Euler–Mascheroni constant world record by Seungmin Kim
- ^ y-cruncher by Alexander Yee
References
- 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
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(help) - 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
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(help) - Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions", INTEGERS: The 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" (PDF). Crelle's Journal (in Latin). 17: 257–285.
{{cite journal}}
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(help) - 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.
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(help) - De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc.
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(help) - 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.
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(help) - Glaisher, James Whitbread Lee (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
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(help) - Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
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(help) - Hardy, G. H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
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(help) - Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
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(help) - 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. doi:10.1090/s0273-0979-2013-01423-x.
{{cite journal}}
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(help) - Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis). Universität des Saarlandes.
- 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.
{{cite journal}}
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(help) - 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. Vol. 71. pp. 219–220. Archived from the original on 2011-06-04. Retrieved 2006-05-29.
{{cite news}}
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(help) - 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.
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(help) 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.
{{cite arXiv}}
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(help) - 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/π and an analog of Hadjicostas's formula", American Mathematical Monthly, 112 (1): 61–65, arXiv:math.CA/0211148, doi:10.2307/30037385, JSTOR 30037385
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at position 70 (help) - Sondow, Jonathan (2005a), New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π, arXiv:math.NT/0508042
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at position 109 (help) - 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.
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(help) - Weisstein, Eric W. (n.d.). "Mertens Constant". mathworld.wolfram.com.
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(help)CS1 maint: year (link) - Yee, Alexander J. (March 7, 2011). "Nagisa - Large Computations". www.numberworld.org.
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(help) - "Records Set by y-cruncher". www.numberworld.org. August 24, 2017. Retrieved April 30, 2018.
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Further reading
- 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.
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(help) 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.
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(help) - Glaisher, James Whitbread Lee (1872). "On the history of Euler's constant". Messenger of Mathematics. 1: 25–30. JFM 03.0130.01.
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(help) - Gourdon, Xavier; Seba, P. (2002). "Collection of formulas for Euler's constant, γ".
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(help) - 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.
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(help) - 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.
{{cite journal}}
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(help) - Knuth, Donald (1997). The Art of Computer Programming, Vol. 1 (3rd ed.). Addison-Wesley. ISBN 0-201-89683-4.
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(help) - Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
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(help) - Mascheroni, Lorenzo (1790), "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur", Galeati, Ticini
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(help) - Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142.
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(help) - 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.
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External links
- "Euler constant", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Weisstein, Eric W. "Euler–Mascheroni constant". MathWorld.
- Jonathan Sondow.
- Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
- Further formulae which make use of the constant: Gourdon and Sebah (2004).