A241011 Decimal expansion of Sierpiński's S~ (S "tilde" as named by S. Finch), a constant appearing in the asymptotics of the number of representations of a positive integer as a sum of two squares.

2, 0, 1, 6, 6, 2, 1, 5, 4, 5, 7, 3, 3, 4, 0, 8, 1, 1, 5, 2, 6, 2, 7, 9, 6, 8, 5, 9, 7, 1, 5, 1, 1, 5, 4, 2, 6, 4, 5, 0, 1, 8, 4, 1, 7, 7, 5, 2, 3, 6, 4, 7, 4, 8, 0, 6, 1, 0, 9, 1, 9, 2, 8, 3, 4, 4, 7, 8, 1, 4, 3, 4, 1, 6, 1, 6, 1, 8, 2, 7, 8, 7, 2, 5, 5, 4, 1, 3, 5, 1, 3, 9, 8, 3, 0, 6, 1, 8, 0, 4

Offset: 1

Author

Jean-François Alcover, Aug 07 2014

Keywords

• nonn
• cons

Comments

From Vaclav Kotesovec, Mar 10 2023: (Start)

Sum_{k=1..n} A002654(k)^2 ~ n * (log(n) + C) / 4, where C = A241011 =

4*gamma - 1 + log(2)/3 - 2*log(Pi) + 8*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.01662154573340811526279685971511542645018417752364748061...

The constant C, published by Ramanujan (1916, formula (22)), 4*gamma - 1 + log(2)/3 - log(Pi) + 4*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.3482276258576268... is wrong! (End)

Examples

			2.01662154573340811526279685971511542645018417752364748061...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's constant, p. 122.

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, Section 2.10 p. 17.
• Srinivasa Ramanujan, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, section (K), formula (22).

Crossrefs

Cf. A062089, A002654, A086058, A241009.

Programs

Mathematica

S = 2*EulerGamma + 2*Log[2] + 3*Log[Pi] - 4* Log[Gamma[1/4]]; (* S~ *) St = 2*S - 12/Pi^2*Zeta'[2] + Log[2]/3 - 1; RealDigits[St, 10, 100] // First

Formula

S_tilde = 2*S - 12/Pi^2*zeta'(2) + log(2)/3 - 1, where S = A086058 - 1 = A062089 / Pi.