Euler's constant

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Euler's constant
Gamma-area.svg
The area of the blue region converges to Euler's constant
Representations
Decimal0.5772156649015328606065120900824024310421...
Continued fraction (linear)[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, ...][1]
Unknown if periodic
Unknown if finite
Binary0.1001001111000100011001111110001101111101...
Hexadecimal0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3A...

Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (γ).

It is defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by

Here, represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:[2]

0.57721566490153286060651209008240243104215933593992...
Unsolved problem in mathematics:

Is Euler's constant irrational? If so, is it transcendental?

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.[3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[5]

Appearances[edit]

Euler's constant appears, among other places, in the following (where '*' means that this entry contains an explicit equation):

Properties[edit]

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.[7][8] The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.[9]

However, some progress was made. Kurt Mahler showed in 1968 that the number is transcendental (here, and are Bessel functions).[10][3] In 2009 Alexander Aptekarev proved that at least one of Euler's constant γ and the Euler–Gompertz constant δ is irrational.[11] This result was improved in 2012 by Tanguy Rivoal, who proved that at least one of them is transcendental.[12][3]

In 2010 M. Ram Murty and N. Saradha considered an infinite list of numbers containing γ/4 and showed that all but at most one of them are transcendental.[3][13] In 2013 M. Ram Murty and A. Zaytseva again considered an infinite list of numbers containing γ and showed that all but at most one are transcendental.[3][14][clarification needed]

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:[15]

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 Euler's constant are the antisymmetric limit:[16]

and the following formula, established in 1898 by de la Vallée-Poussin:

where are ceiling brackets. This formula indicates that when taking any positive integer n and dividing it by each positive integer k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to (rather than 0.5) as n tends to infinity.

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[edit]

γ equals the value of a number of definite integrals:

where Hx is the fractional harmonic number.

The third formula in the integral list can be proved in the following way:

The integral on the second line of the equation stands for the Debye function value of +∞, which is m! ζ(m + 1).

Definite integrals in which γ appears include:

One can express γ using a special case of Hadjicostas's formula as a double integral[9][17] with equivalent series:

An interesting comparison by Sondow[17] is the double integral and alternating series

It shows that log 4/π may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series[18]

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[19]

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 .[20][21] 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:[15][22]

In 1910, Vacca found the closely related series[23][24][25][26][27][15][28]

where log2 is the logarithm to base 2 and ⌊ ⌋ is the floor function.

In 1926 he found a second series:

From the MalmstenKummer expansion for the logarithm of the gamma function[29] we get:

An important expansion for Euler's constant is due to Fontana and Mascheroni

where Gn are Gregory coefficients[15][28][30] This series is the special case of the expansions

convergent for

A similar series with the Cauchy numbers of the second kind Cn is[28][31]

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[32][33]

Other series with the same polynomials include these examples:

and

where Γ(a) is the gamma function.[30]

A series related to the Akiyama-Tanigawa algorithm is

where Gn(2) are the Gregory coefficients of the second order.[30]

Series of prime numbers:

Asymptotic expansions[edit]

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 derived closed-form expressions for the sums of errors of these approximations.[31] He showed that (Theorem A.1):

Exponential[edit]

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.[34] The numerical value of eγ is:[35]

1.78107241799019798523650410310717954916964521430343....

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 using hypergeometric functions.[36]

It also holds that[37]

Continued fraction[edit]

The continued fraction expansion of γ begins [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...],[1] which has no apparent pattern. The continued fraction is known to have at least 475,006 terms,[7] and it has infinitely many terms if and only if γ is irrational.

Generalizations[edit]

abm(x) = γx

Euler's generalized constants are given by

for 0 < α < 1, with γ as the special case α = 1.[38] 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:[13]

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.

Published Decimal Expansions of γ
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 [39]
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 [40][41]
December 22, 2013 119377958182 Alexander J. Yee [41]
March 15, 2016 160000000000 Peter Trueb [41]
May 18, 2016 250000000000 Ron Watkins [41]
August 23, 2017 477511832674 Ron Watkins [41]
May 26, 2020 600000000100 Seungmin Kim & Ian Cutress [41][42]

References[edit]

  • Bretschneider, Carl Anton (1837) [1835]. "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal (in Latin). 17: 257–285.
  • Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
  • Ram Murty, M.; Saradha, N. (2010). "Euler–Lehmer constants and a conjecture of Erdos". Journal of Number Theory. 130 (12): 2671–2681. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.

Footnotes

  1. ^ a b Sloane, N. J. A. (ed.). "Sequence A002852 (Continued fraction for Euler's constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A001620 (Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ a b c d e 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. S2CID 119612431.
  4. ^ Bretschneider 1837, "γ = c = 0,577215 664901 532860 618112 090082 3.." on p. 260.
  5. ^ De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc. "γ" on p. 578.
  6. ^ 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.
  7. ^ 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. 1423: 338–350. doi:10.1007/bfb0054873. ISBN 9783540691136.
  8. ^ Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis) (in German). Universität des Saarlandes.
  9. ^ a b See also Sondow, Jonathan (2003). "Criteria for irrationality of Euler's constant". Proceedings of the American Mathematical Society. 131 (11): 3335–3344. arXiv:math.NT/0209070. doi:10.1090/S0002-9939-03-07081-3. S2CID 91176597.
  10. ^ Mahler, Kurt; Mordell, Louis Joel (4 June 1968). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
  11. ^ Aptekarev, A. I. (28 February 2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
  12. ^ Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
  13. ^ a b Ram Murty & Saradha 2010.
  14. ^ Murty, M. Ram; Zaytseva, Anastasia (2013). "Transcendence of Generalized Euler Constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890. JSTOR 10.4169/amer.math.monthly.120.01.048. S2CID 20495981.
  15. ^ a b c d Krämer, Stefan (2005). Die Eulersche Konstante γ und verwandte Zahlen (in German). University of Göttingen.
  16. ^ Sondow, Jonathan (1998). "An antisymmetric formula for Euler's constant". Mathematics Magazine. 71 (3): 219–220. doi:10.1080/0025570X.1998.11996638. Archived from the original on 2011-06-04. Retrieved 2006-05-29.
  17. ^ a b Sondow, Jonathan (2005), "Double integrals for Euler's constant and and an analog of Hadjicostas's formula", American Mathematical Monthly, 112 (1): 61–65, arXiv:math.CA/0211148, doi:10.2307/30037385, JSTOR 30037385
  18. ^ Sondow, Jonathan (1 August 2005a). New Vacca-type rational series for Euler's constant and its 'alternating' analog . arXiv:math.NT/0508042.
  19. ^ 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. S2CID 1368088.
  20. ^ 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.
  21. ^ Havil 2003, pp. 75–8.
  22. ^ Blagouchine 2016.
  23. ^ Vacca, G. (1910). "A new analytical expression for the number π and some historical considerations". Bulletin of the American Mathematical Society. 16: 368–369. doi:10.1090/S0002-9904-1910-01919-4.
  24. ^ Glaisher, James Whitbread Lee (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
  25. ^ Hardy, G.H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
  26. ^ 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 (in Italian). 6 (3): 19–20.
  27. ^ Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
  28. ^ a b c 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
  29. ^ 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. S2CID 120943474.
  30. ^ a b c 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
  31. ^ a b Alabdulmohsin, Ibrahim M. (2018). Summability Calculus. A Comprehensive Theory of Fractional Finite Sums. Springer. pp. 147–8. ISBN 9783319746487.
  32. ^ Sloane, N. J. A. (ed.). "Sequence A302120 (Absolute value of the numerators of a series converging to Euler's constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  33. ^ Sloane, N. J. A. (ed.). "Sequence A302121 (Denominators of a series converging to Euler's constant)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  34. ^ Weisstein, Eric W. "Mertens Constant". WolframMathWorld (Wolfram Research).
  35. ^ Sloane, N. J. A. (ed.). "Sequence A073004 (Decimal expansion of exp(gamma))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  36. ^ Sondow, Jonathan (2003). "An infinite product for eγ via hypergeometric formulas for Euler's constant, γ". arXiv:math.CA/0306008.
  37. ^ Choi, Junesang; Srivastava, H.M. (1 September 2010). "Integral Representations for the Euler–Mascheroni Constant γ". Integral Transforms and Special Functions. 21 (9): 675–690. doi:10.1080/10652461003593294. ISSN 1065-2469. S2CID 123698377.
  38. ^ Havil 2003, pp. 117–8.
  39. ^ Knuth, Donald E. (July 1962). "Euler's Constant to 1271 Places". Mathematics of Computation. American Mathematical Society. 16 (79): 275–281. JSTOR 2004048.
  40. ^ Yee, Alexander J. (7 March 2011). "Large Computations". www.numberworld.org.
  41. ^ a b c d e f Yee, Alexander J. "Records Set by y-cruncher". www.numberworld.org. Retrieved 30 April 2018.
    Yee, Alexander J. "y-cruncher - A Multi-Threaded Pi-Program". www.numberworld.org.
  42. ^ "Euler-Mascheroni Constant". Polymath Collector. 15 February 2020.

Further reading[edit]

External links[edit]