A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.
5, 7, 7, 2, 1, 5, 6, 6, 4, 9, 0, 1, 5, 3, 2, 8, 6, 0, 6, 0, 6, 5, 1, 2, 0, 9, 0, 0, 8, 2, 4, 0, 2, 4, 3, 1, 0, 4, 2, 1, 5, 9, 3, 3, 5, 9, 3, 9, 9, 2, 3, 5, 9, 8, 8, 0, 5, 7, 6, 7, 2, 3, 4, 8, 8, 4, 8, 6, 7, 7, 2, 6, 7, 7, 7, 6, 6, 4, 6, 7, 0, 9, 3, 6, 9, 4, 7, 0, 6, 3, 2, 9, 1, 7, 4, 6, 7, 4, 9
Offset: 0
Examples
0.577215664901532860606512090082402431042...
References
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 3.
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 259-262.
- S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 28-40, 166, 365.
- C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.
- B. Gugger, Problèmes corrigés de Mathématiques posés aux concours des Ecoles Militaires, Ecole de l'Air, 1992, option MP, 1ère épreuve, Ellipses, 1993, pp. 167-184.
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.3 Infinite Series, pp. 273-274.
- J. Havil, Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.
- J.-M. Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, Exercice 4.3.14, pages 371 and 387, 1997.
- Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 166.
- Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 44.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equation 1:7:5 at page 13.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 28.
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1990.
Links
- Harry J. Smith, Table of n, a(n) for n = 0..20000
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/Pi^2 and into the formal enveloping series with rational coefficients only, arXiv:1501.00740 [math.NT], 2015-2016; Journal of Number Theory (Elsevier), Volume 158, pages 365-396, 2016.
- D. Bradley, Ramanujan's formula for the logarithmic derivative of the Gamma function, arXiv:math/0505125 [math.CA], 2005.
- R. P. Brent, Ramanujan and Euler's constant.
- R. P. Brent and F. Johansson, A bound for the error term in the Brent-McMillan algorithm, arXiv 1312.0039 [math.NA], Nov. 2013.
- C. K. Caldwell, The Prime Glossary, Euler's constant.
- D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.
- Chao-Ping Chen, Sharp inequalities and asymptotic series of a product related to the Euler-Mascheroni constant, Journal of Number Theory, Volume 165, August 2016, Pages 314-323.
- E. Chlebus, A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant, Amer. Math. Mnthly, 118 (2011), 268-274.
- M. Coffey and Jonathan Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, arXiv:1202.3093 [math.NT], 2012; Acta Appl. Math., 121 (2012), 1-3.
- Dave's Math Tables, Gamma Constant.
- Philippe Deléham, Letter to N. J. A. Sloane, Apr 14 1997.
- Thomas and Joseph Dence, A survey of Euler's constant, Math. Mag., 82 (2009), 255-265.
- Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016.
- Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants.
- X. Gourdon and P. Sebah, The Euler constant gamma.
- Kalpok Guha and Sourangshu Ghosh, Measuring Abundance with Abundancy Index, (2021).
- Brady Haran and Tony Padilla, The mystery of 0.577, Numberphile video, 2016.
- J. C. Kluyver, Euler's constant and natural numbers, Proc. K. Ned. Akad. Wet., 27(1-2) (1924), 142-144.
- D. E. Knuth, Euler's constant to 1271 places, Math. Comp. 16 1962 275-281.
- Stefan Krämer, Euler's Constant γ=0.577... Its Mathematics and History.
- Richard Kreckel, 116 million digits of Euler's constant (bzipped).
- A. Krowne, PlanetMath.org, Euler's constant.
- Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.
- M. Lerch, Expressions nouvelles de la constante d'Euler, S.-B. Kgl. Bohmischen Ges. Wiss., Article XLII (1897), Prague (5 pages).
- Eric Naslund, Euler-Mascheroni constant expression, further simplification, MathStackExchange.
- T. Papanikolaou, Plouffe's Inverter, Euler's constant to 1000000 decimals.
- Michael Penn, Euler's other constant, YouTube video (2023).
- S. Plouffe, using data from J. Borwein, 170000 digits of Euler or gamma constant [archived copy of a page on WorldWideSchool.org which doesn't exist any mode, cf. "Euler" link in left column].
- S. Ramanujan, A series for Euler's constant, Messenger of Math., 46 (1917), 73-80.
- S. Ramanujan, Question 327, J. Ind. Math. Soc.
- Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
- Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
- Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
- Jonathan Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma, arXiv:math/0306008 [math.CA], 2003.
- Jonathan Sondow, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin, arXiv:math/0211075 [math.NT], 2002-2009; Math. Slovaca 59 (2009), 1-8.
- Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
- Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006; J. Math. Anal. Appl. 332 (1) (2007), 292-314.
- Jonathan Sondow and S. Zlobin, Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant, arXiv:0705.0732 [math.NT], 2007; Math. Notes, 84 (2008), 568-583, Erratum p. 887.
- Jonathan Sondow and W. Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, arXiv:math/0304021 [math.NT], 2003; Ramanujan J. 12 (2006), 225-244.
- D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.
- Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
- Wikipedia, Stieltjes constants.
- A. Y. Yee, Large computations.
- Index entries for sequences related to Beatty sequences.
Crossrefs
Programs
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Magma
EulerGamma(250); // G. C. Greubel, Aug 21 2018
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Maple
Digits := 100; evalf(gamma);
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Mathematica
RealDigits[ EulerGamma, 10, 105][[1]] (* Robert G. Wilson v, Nov 01 2004 *) (1/2) N[Sum[PolyGamma[0, 1/2 + 2^k] - PolyGamma[0, 2^k], {k, 0, Infinity }], 30] (* Dimitri Papadopoulos, Nov 30 2016 *)
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PARI
default(realprecision, 20080); x=Euler; d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b001620.txt", n, " ", d)); \\ Harry J. Smith, Apr 15 2009 -
Python
from sympy import S def aupton(digs): return [int(d) for d in str(S.EulerGamma.n(digs+2))[2:-2]] print(aupton(99)) # Michael S. Branicky, Nov 22 2021
Formula
Limit_{n->oo} (1 + 1/2 + ... + 1/n - log(n)) (definition).
Sum_{n>=1} (1/n - log(1 + 1/n)), since log(1 + 1/1) + ... + log(1 + 1/n) telescopes to log(n+1) and lim_{n->infinity} (log(n+1) - log(n)) = 0.
Integral_{x=0..1} -log(log(1/x)). - Robert G. Wilson v, Jan 04 2006
Integral_{x=0..1,y=0..1} (x-1)/((1-x*y)*log(x*y)). - (see Sondow 2005)
Integral_{x=0..oo} -log(x)*exp(-x). - Jean-François Alcover, Mar 22 2013
Integral_{x=0..1} (1 - exp(-x) - exp(-1/x))/x. - Jean-François Alcover, Apr 11 2013
Equals the lim_{n->oo} fractional part of zeta(1+1/n). The corresponding fractional part for x->1 from below, using n-1/n, is -(1-a(n)). The fractional part found in this way for the first derivative of Zeta as x->1 is A252898. - Richard R. Forberg, Dec 24 2014
Limit_{x->1} (Zeta(x)-1/(x-1)) from Whittaker and Watson. 1990. - Richard R. Forberg, Dec 30 2014
exp(gamma) = lim_{i->oo} exp(H(i)) - exp(H(i-1)), where H(i) = i-th Harmonic number. For a given n this converges faster than the standard definition, and two above, after taking the logarithm (e.g., 13 digits vs. 6 digits at n=3000000 or x=1+1/3000000). - Richard R. Forberg, Jan 08 2015
Limit_{n->oo} (1/2) Sum_{j>=1} Sum_{k=1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n) (k + j*n))). - Dimitri Papadopoulos, Jan 13 2016
Equals 25/27 minus lim_{x->oo} 2^(x+1)/3 - (22/27)*(4/3)^x - Zeta(Sum_{i>=1} (H_i/i^x)), letting H_i denote the i-th harmonic number. - John M. Campbell, Jan 29 2016
Limit_{x->0} -B'(x), where B(x) = -x zeta(1-x) is the "Bernoulli function". - Jean-François Alcover, May 20 2016
Sum_{k>=0} (1/2)(digamma(1/2+2^k) - digamma(2^k)) where digamma(x) = d/dx log(Gamma(x)). - Dimitri Papadopoulos, Nov 14 2016
Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma = -Pi*Integral_{0..oo} a/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..oo} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k) *a^(n-2*k). - Peter Luschny, Apr 19 2018
Limit_{s->0} (Zeta'(1-s)*s - Zeta(1-s)) / (Zeta(1-s)*s). - Peter Luschny, Jun 18 2018
log(2) * (gamma - (1/2) * log(2)) = -Sum_{v >= 1} (1/2^(v+1)) * (Delta^v (log(w)/w))|{w=1}, where Delta(f(w)) = f(w) - f(w + 1) (forward difference). [This is a formula from Lerch (1897).] - _Petros Hadjicostas, Jul 21 2019
From Amiram Eldar, Jul 05 2020: (Start)
Equals Integral_{x=1..oo} (1/floor(x) - 1/x) dx.
Equals Integral_{x=0..1} (1/(1-x) + 1/log(x)) dx = Integral_{x=0..1} (1/x + 1/log(1-x)) dx.
Equals -Integral_{-oo..oo} x*exp(x-exp(x)) dx.
Equals Sum_{k>=1} (-1)^k * floor(log_2(k))/k.
Equals (-1/2) * Sum_{k>=1} (Lambda(k)-1)/k, where Lambda is the Mangoldt function. (End)
Equals Integral_{0..1} -1/LambertW(-1,-x*exp(-x)) dx = 1 + Integral_{0..1} LambertW(-1/x*exp(-1/x)) dx. - Gleb Koloskov, Jun 12 2021
Equals Sum_{k>=2} (-1)^k * zeta(k)/k. - Vaclav Kotesovec, Jun 19 2021
Equals lim_{x->oo} log(x) - Sum_{p prime <= x} log(p)/(p-1). - Amiram Eldar, Jun 29 2021
Limit_{n->oo} (2*HarmonicNumber(n) - HarmonicNumber(n^2)). After answer by Eric Naslund on Mathematics Stack Exchange, on Jun 21 2011. - Mats Granvik, Jul 19 2021
Equals Integral_{x=0..oo} ( exp(-x) * (1/(1-exp(-x)) - 1/x) ) dx (see Gugger or Monier). - Bernard Schott, Nov 21 2021
Equals 1/2 + Limit_{s->1} (Zeta(s) + Zeta(1/s))/2. - Thomas Ordowski, Jan 12 2023
Equals Sum_{j>=2} Sum_{k>=2} ((k-1)/(k*j^k)). - Mike Tryczak, Apr 06 2023
From Stefano Spezia, Oct 27 2024: (Start)
Equals Sum_{n>=1} n*(zeta(n+1) - 1)/(n + 1) [Euler] (see Finch at p. 30).
Equals lim_{n->oo} Sum_{prime p<=n} log(p/(p - 1)) - log(log(n)) (see Finch at p. 31). (End)
Equals lim_{s->1} zeta(s) - zeta(s)^2/zeta(2*s - 1)/2. - Mats Granvik, Jul 07 2025
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