Euler–Mascheroni constant
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, ...]^{[1]} (It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic. Shown in linear notation) |
Contents
History[edit]
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.^{[2]} For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835^{[3]} and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.^{[4]}
Appearances[edit]
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^{[5]}
Properties[edit]
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 10^{242080}.^{[6]} 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[edit]
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[edit]
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, H_{n}. Expanding some of the terms in the Hurwitz zeta function gives:
where 0 < ε < 1/252n^{6}.
Integrals[edit]
γ equals the value of a number of definite integrals:
where H_{x} 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, 2005) with equivalent series:
An interesting comparison by J. 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 (see Sondow 2005 #2)
where N_{1}(n) and N_{0}(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)
Series expansions[edit]
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 .^{[7]}^{[8]} 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:^{[9]}
In 1910, Vacca found the closely related series^{[10]}
where log_{2} 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^{[11]} we get:
An important expansion for Euler's constant is due to Fontana and Mascheroni
where G_{n} are Gregory coefficients.^{[12]} This series is the special case of the expansions
convergent for
A similar series with the Cauchy numbers of the second kind C_{n} is^{[13]}
Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series
where ψ_{n}(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 OEIS: A302120 and OEIS: A302121. Other series with the same polynomials include these examples:
and
where Γ(a) is the gamma function.^{[14]}
A series related to the Akiyama-Tanigawa algorithm is
where G_{n}(2) are the Gregory coefficients of the second order.^{[14]}
Series of prime numbers:
Asymptotic expansions[edit]
γ equals the following asymptotic formulas (where H_{n} is the nth harmonic number):
- (Euler)
- (Negoi)
- (Cesàro)
The third formula is also called the Ramanujan expansion.
Exponential[edit]
The constant e^{γ} is important in number theory. Some authors denote this quantity simply as γ′. e^{γ} equals the following limit, where p_{n} is the nth prime number:
This restates the third of Mertens' theorems.^{[15]} The numerical value of e^{γ} is:
Other infinite products relating to e^{γ} 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.
Continued fraction[edit]
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,^{[6]} and it has infinitely many terms if and only if γ is irrational.
Generalizations[edit]
Euler's generalized constants are given by
for 0 < α < 1, with γ as the special case α = 1.^{[16]} 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:^{[17]}^{[18]}
The basic properties are
and if gcd(a,q) = d then
Published digits[edit]
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 |
---|---|---|
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 | 050 1 | Helmut Fischer and Karl Zeller |
1962 | 271 1 | Donald Knuth |
1962 | 566 3 | Dura W. Sweeney |
1973 | 879 4 | William A. Beyer and Michael S. Waterman |
1977 | 700 20 | Richard P. Brent |
1980 | 100 30 | Richard P. Brent & Edwin M. McMillan |
1993 | 000 172 | Jonathan Borwein |
1999 | 000000 108 | Patrick Demichel and Xavier Gourdon |
2009 | 844489545 29 | Alexander J. Yee & Raymond Chan^{[19]}^{[20]} |
2013 | 377958182 119 | Alexander J. Yee^{[19]}^{[20]} |
2016 | 000000000 160 | Peter Trueb^{[20]} |
2016 | 000000000 250 | Ron Watkins^{[20]} |
2017 | 511832674 477 | Ron Watkins^{[20]} |
Notes[edit]
- Footnotes
- ^ OEIS: A002852
- ^ Lagarias, Jeffrey C. (October 2013). "Euler's constant: Euler's work and modern developments" (PDF). Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. Bibcode:1994BAMaS..30..205W. 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)
- ^ 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.
- ^ ^{a} ^{b} Havil (2003) p. 97.
- ^ 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.
- ^ Havil (2003), pp. 75-78
- ^ Krämer (2005), Blagouchine (2016).
- ^ Vacca (1910), Glaisher (1910), Hardy (1912), Vacca (1925), Kluyver (1927), Krämer (2005), Blagouchine (2016).
- ^ Blagouchine (2014).
- ^ Krämer (2005), Blagouchine (2016), Blagouchine (2018).
- ^ Blagouchine (2016), Alabdulmohsin (2018), pp. 147-148.
- ^ ^{a} ^{b} Blagouchine (2018).
- ^ http://mathworld.wolfram.com/MertensConstant.html (14)
- ^ Havil (2003), pp. 117–118
- ^ 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.
- ^ Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142.
- ^ ^{a} ^{b} "Nagisa - Large Computations". www.numberworld.org.
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} "Records Set by y-cruncher". August 24, 2017. Retrieved April 30, 2018.
- 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", 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, 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, Bibcode:2016arXiv160602044B arXiv
- 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.
- Carl Anton, Bretschneider (1837). "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal. 17: 257–285. (submitted 1835)
- Gerst, I. (1969). "Some series for Euler's constant". Amer. Math. Monthly. 76 (3): 237–275. doi:10.2307/2316370. JSTOR 2316370.
- James Whitbread Lee Glaisher (1872). "On the history of Euler's constant". Messenger of Mathematics. 1: 25–30., JFM 03.0130.01
- James Whitbread Lee Glaisher (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
- Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ."
- Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ."
- 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.
- 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.
- Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
- 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.
- 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.
- Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
- Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.
- 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:math.NT/0211075. Bibcode:2002math.....11075S. with an Appendix by Sergey Zlobin
- Sondow, Jonathan (2003). "An infinite product for e^{γ} 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. pp. 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.
- Sondow, Jonathan (2005). "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/π". 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. 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.
External links[edit]
- 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).