|
%I #32 Jun 13 2021 05:36:36
%S 8,2,2,8,2,5,2,4,9,6,7,8,8,4,7,0,3,2,9,9,5,3,2,8,7,1,6,2,6,1,4,6,4,9,
%T 4,9,4,7,5,6,9,3,1,1,8,8,9,4,8,5,0,2,1,8,3,9,3,8,1,5,6,1,3,0,3,7,0,9,
%U 0,9,5,6,4,4,6,4,0,1,6,6,7,5,7,2,1,9,5,3,2,5,7,3,2,3,4,4,5,3,2,4,7,2,1,4
%N Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.
%H M. W. Coffey, <a href="https://arxiv.org/abs/1701.07064">Summatory relations and prime products for the Stieltjes constants, and other related results</a>, arXiv:1701.07064 (2017) Proposition 9.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
%H Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, <a href="http://www.ams.org/journals/mcom/2013-82-282/S0025-5718-2012-02635-4/S0025-5718-2012-02635-4.pdf">Numerical approximation of the Masser-Gramain constant to four decimal places</a>, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/SierpinskiConstant.html">Sierpiński's Constant</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpi%C5%84ski's_constant">Sierpiński's Constant</a>
%F S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function.
%F S = log[Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant.
%F S = log(4*Pi^3*exp(2*gamma) / Gamma(1/4)^4), where gamma is Euler's constant and Gamma is Euler's Gamma function.
%F S = A062089 / Pi, where A062089 is Sierpiński's K constant.
%F S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by _Vaclav Kotesovec_, Apr 27 2015]
%F S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx.
%F Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)).
%F Equals 2*A001620 - A088538*A115252 [Coffey]. - _R. J. Mathar_, Jan 15 2021
%e 0.822825249678847032995328716261464949475693118894850218393815613...
%t S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First
%Y Cf. A062089, A062539, A068466, A086058.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Aug 08 2014
|
