A319091 Decimal expansion of D, the coefficient of n in the asymptotic formula of Ramanujan for Sum_{k=1..n} (d(k)^2), where d(k) is the number of distinct divisors of k.
4, 6, 0, 3, 2, 3, 3, 7, 2, 2, 5, 8, 7, 2, 1, 4, 3, 0, 3, 9, 3, 7, 6, 2, 0, 8, 6, 3, 8, 4, 4, 1, 8, 9, 7, 4, 7, 6, 3, 2, 1, 4, 9, 0, 3, 5, 3, 8, 7, 3, 9, 2, 2, 4, 0, 5, 8, 4, 2, 5, 0, 3, 4, 8, 4, 4, 5, 9, 0, 2, 6, 2, 9, 3, 2, 4, 0, 3, 2, 0, 7, 3, 8, 0, 1, 9, 8, 4, 8, 1, 0, 7, 6, 5, 9, 8, 5, 9, 9, 7, 3, 5, 6, 9, 5, 8
Offset: 0
Examples
0.4603233722587214303937620863844189747632149035387392240584250348445902629324...
Links
- Ramanujan's Papers, Some formulas in the analytic theory of numbers Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).
Programs
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Mathematica
24*EulerGamma^3/Pi^2 - (432*Zeta'[2] /Pi^4+ 36/Pi^2)*EulerGamma^2 + (3456*Zeta'[2]^2/Pi^6 + 288*(Zeta'[2]-Zeta''[2])/Pi^4 + 24/Pi^2 - 72*StieltjesGamma[1]/Pi^2)*EulerGamma + StieltjesGamma[1]*(288*Zeta'[2]/Pi^4 + 24/Pi^2)-10368*Zeta'[2]^3/Pi^8 - 864*Zeta'[2]^2/Pi^6 + 1728*Zeta''[2] * Zeta'[2]/Pi^6 + 72*(Zeta''[2]-Zeta'[2])/Pi^4 - 48*Zeta'''[2]/Pi^4 + (12*StieltjesGamma[2] - 6)/Pi^2
Formula
D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279.