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A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.
(Formerly M3755 N1532)

%I M3755 N1532

%S 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,

%T 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,

%U 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

%N Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.

%C Yee (2010) computed 29844489545 decimal digits of gamma.

%C Decimal expansion of 0th Stieltjes constant. - _Paul Muljadi_, Aug 24 2010

%C The value of Euler's constant is close to (18/Pi^2)*Sum_{n>=0} 1/4^(2^n) = 0.5770836328... = (6/5) * A082020 * A078585. - _Arkadiusz Wesolowski_, Mar 27 2012

%D 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.

%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 28-40.

%D C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.

%D J. Havil, Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1990.

%H Harry J. Smith, <a href="/A001620/b001620.txt">Table of n, a(n) for n = 0..20000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Iaroslav V. Blagouchine, <a href="http://arxiv.org/abs/1501.00740">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</a>, arXiv:1501.00740 [math.NT], 2015-2016; Journal of Number Theory (Elsevier), Volume 158, pages 365-396, 2016.

%H D. Bradley, <a href="http://arXiv.org/abs/math.CA/0505125">Ramanujan's formula for the logarithmic derivative of the Gamma function</a>, arXiv:math/0505125 [math.CA], 2005.

%H R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub139.html">Ramanujan and Euler's constant</a>

%H R. P. Brent and F. Johansson,<a href="http://arxiv.org/abs/1312.0039">A bound for the error term in the Brent-McMillan algorithm</a>, arXiv 1312.0039 [math.NA], Nov. 2013.

%H C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=Gamma">Euler's constant</a>

%H D. Castellanos, <a href="http://www.jstor.org/stable/2690037">The ubiquitous pi</a>, Math. Mag., 61 (1988), 67-98 and 148-163.

%H Chao-Ping Chen, <a href="https://doi.org/10.1016/j.jnt.2016.01.021">Sharp inequalities and asymptotic series of a product related to the Euler-Mascheroni constant</a>, Journal of Number Theory, Volume 165, August 2016, Pages 314-323.

%H E. Chlebus, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.268">A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant</a>, Amer. Math. Mnthly, 118 (2011), 268-274.

%H M. Coffey and J. Sondow, <a href="http://arxiv.org/abs/1202.3093">Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant</a>, arXiv:1202.3093 [math.NT], 2012; Acta Appl. Math., 121 (2012), 1-3.

%H Dave's Math Tables, <a href="http://math2.org/math/constants/gamma.htm">Gamma Constant</a>

%H Philippe Deléham, <a href="/A009763/a009763.pdf">Letter to N. J. A. Sloane, Apr 14 1997</a>

%H Thomas and Joseph Dence, <a href="https://www.jstor.org/stable/27765916">A survey of Euler's constant</a>, Math. Mag., 82 (2009), 255-265.

%H Pierre Dusart, <a href="https://doi.org/10.1007/s11139-016-9839-4">Explicit estimates of some functions over primes</a>, The Ramanujan Journal, 2016.

%H Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>

%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Gamma/gamma.html">The Euler's constant gamma</a>

%H Brady Haran and Tony Padilla, <a href="https://www.youtube.com/watch?v=4k1jegU4Wb4">The mystery of 0.577</a>, Numberphile video, 2016.

%H J. C. Kluyver, <a href="http://www.dwc.knaw.nl/DL/publications/PU00015025.pdf">Euler's constant and natural numbers</a>, Proc. K. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

%H D. E. Knuth, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0148255-X">Euler's constant to 1271 places</a>, Math. Comp. 16 1962 275-281.

%H Stefan Krämer, <a href="http://www.math.uni-goettingen.de/skraemer/gamma.html">Euler's Constant γ=0.577... Its Mathematics and History</a>

%H Richard Kreckel, <a href="http://www.ginac.de/~kreckel/news.html">116 million digits of Euler's constant</a> (bzipped)

%H A. Krowne, PlanetMath.org, <a href="http://planetmath.org/eulersconstant">Euler's constant</a>

%H Jeffrey C. Lagarias, <a href="http://arxiv.org/abs/1303.1856">Euler's constant: Euler's work and modern developments</a>, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.

%H M. Lerch, <a href="https://www.zobodat.at/pdf/SB-Ges-Wiss-Prag_1896_2_0001-0687.pdf">Expressions nouvelles de la constante d'Euler</a>, S.-B. Kgl. Bohmischen Ges. Wiss., Article XLII (1897), Prague (5 pages).

%H T. Papanikolaou, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/gamma.txt">Euler's constant to 1000000 decimals</a>

%H S. Plouffe, using data from J. Borwein, <a href="https://archive.is/EYc1c">170000 digits of Euler or gamma constant</a> [archived copy of a page on WorldWideSchool.org which doesn't exist any mode, cf. "Euler" link in left column].

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper19/page1.htm">A series for Euler's constant</a>, Messenger of Math., 46 (1917), 73-80.

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q327.htm">Question 327</a>, J. Ind. Math. Soc.

%H J. Sondow, <a href="https://www.jstor.org/stable/2691211">An antisymmetric formula for Euler's constant</a>, Math. Mag. 71 (1998), 219-220.

%H J. Sondow, <a href="https://doi.org/10.1090/S0002-9939-03-07081-3">Criteria for irrationality of Euler's constant</a>, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.

%H J. Sondow, <a href="https://arxiv.org/abs/math/0211148">Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.

%H J. Sondow, <a href="https://arxiv.org/abs/math/0306008">An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma</a>, arXiv:math/0306008 [math.CA], 2003.

%H J. Sondow, <a href="https://arxiv.org/abs/math/0211075">A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin</a>, arXiv:math/0211075 [math.NT], 2002-2009; Math. Slovaca 59 (2009), 1-8.

%H J. Sondow, <a href="https://arxiv.org/abs/math/0508042">New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)</a>, 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.

%H J. Sondow and P. Hadjicostas, <a href="http://arXiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, arXiv:math/0610499 [math.CA], 2006; J. Math. Anal. Appl. 332 (1) (2007), 292-314.

%H J. Sondow and S. Zlobin, <a href="http://arxiv.org/abs/0705.0732">Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant</a>, arXiv:0705.0732 [math.NT], 2007; Math. Notes, 84 (2008), 568-583, Erratum p. 887.

%H J. Sondow and W. Zudilin, <a href="https://arxiv.org/abs/math/0304021">Euler's constant, q-logarithms and formulas of Ramanujan and Gosper</a>, arXiv:math/0304021 [math.NT], 2003; Ramanujan J. 12 (2006), 225-244.

%H D. W. Sweeney, <a href="http://dx.doi.org/10.1090/S0025-5718-1963-0160308-X">On the computation of Euler's constant</a>, Math. Comp., 17 (1963), 170-178.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes constants</a>

%H A. Y. Yee, <a href="http://www.numberworld.org/nagisa_runs/computations.html">Large computations</a>

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F Lim_{n->infinity} (1 + 1/2 + ... + 1/n - log(n)) (definition).

%F 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.

%F Integral_{x=0..1} -log(log(1/x)). - _Robert G. Wilson v_, Jan 04 2006

%F Integral_{x=0..1,y=0..1} (x-1)/((1-x*y)*log(x*y)). - (see Sondow 2005)

%F Integral_{x=0..infinity} -log(x)*exp(-x). - _Jean-François Alcover_, Mar 22 2013

%F Integral_{x=0..1} (1 - exp(-x) - exp(-1/x))/x. - _Jean-François Alcover_, Apr 11 2013

%F Equals the lim_{n->infinity} 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

%F Lim_{x->1} (Zeta(x)-1/(x-1)) from Whittaker and Watson. 1990. - _Richard R. Forberg_, Dec 30 2014

%F exp(gamma) = lim_{i->infinity} 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

%F Lim_{n->infinity} (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

%F Equals 25/27 minus lim_{x->infty} 2^(x+1)/3-22/27*(4/3)^x-Zeta(sum(H_i/i^x,i=1..infty)), letting H_i denote the i-th harmonic number. - _John M. Campbell_, Jan 29 2016

%F Lim_{x->0} -B'(x), where B(x) = -x zeta(1-x) is the "Bernoulli function". - _Jean-François Alcover_, May 20 2016

%F 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

%F a(n) = -10*floor(gamma*10^n) + floor(gamma*10^(n + 1)) for n >= 0. - _Mariusz Iwaniuk_, Apr 28 2017

%F Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma = -Pi*Integral_{0..infinity} 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..infinity} 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

%F Lim_{s->0} (Zeta'(1-s)*s - Zeta(1-s)) / (Zeta(1-s)*s). - _Peter Luschny_, Jun 18 2018

%F 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

%F From _Amiram Eldar_, Jul 05 2020: (Start)

%F Equals Integral_{x=1..oo} (1/floor(x) - 1/x) dx.

%F Equals Integral_{x=0..1} (1/(1-x) + 1/log(x)) dx = Integral_{x=0..1} (1/x + 1/log(1-x)) dx.

%F Equals -Integral_{-oo..oo} x*exp(x-exp(x)) dx.

%F Equals Sum_{k>=1} (-1)^k * floor(log2(k))/k.

%F Equals (-1/2) * Sum_{k>=1} (Lambda(k)-1)/k, where Lambda is the Mangoldt function. (End)

%e 0.577215664901532860606512090082402431042...

%p Digits := 100; evalf(gamma);

%t RealDigits[ EulerGamma, 10, 105][[1]] (* _Robert G. Wilson v_, Nov 01 2004 *)

%t (1/2) N[Sum[PolyGamma[0, 1/2 + 2^k] - PolyGamma[0, 2^k], {k, 0, Infinity }], 30] (* _Dimitri Papadopoulos_, Nov 30 2016 *)

%o (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

%o (MAGMA) EulerGamma(250); // _G. C. Greubel_, Aug 21 2018

%Y Cf. A002852 (continued fraction).

%Y Cf. A073004 (exp(gamma)) and A094640 ("alternating Euler constant").

%Y Cf. A231095 (power tower using this constant).

%Y Cf. A199332, A252898.

%Y Denote the generalized Euler constants, also called Stieltjes constants, by Sti(n).

%Y Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),

%Y Sti(1/2) = A301816, Sti(1) = A082633 (cf. A262382/A262383), Sti(3/2) = A301817,

%Y Sti(2) = A086279 (cf. A262384/A262385), Sti(3) = A086280 (cf. A262386/A262387),

%Y Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,

%Y Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

%K nonn,cons,nice

%O 0,1

%A _N. J. A. Sloane_

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