A241017 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.
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, 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, 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
Offset: 0
Examples
0.822825249678847032995328716261464949475693118894850218393815613...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.
Links
- M. W. Coffey, Summatory relations and prime products for the Stieltjes constants, and other related results, arXiv:1701.07064 (2017) Proposition 9.
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
- Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, Numerical approximation of the Masser-Gramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246
- Eric Weisstein's MathWorld, Sierpiński's Constant
- Wikipedia, Sierpiński's Constant
Programs
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Mathematica
S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First
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PARI
log(agm(sqrt(2), 1)^2/2) + 2*Euler \\ Charles R Greathouse IV, Nov 26 2024
Formula
S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function.
S = log(Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant.
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.
S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015]
S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx.
Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)).