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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.
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%I #7 Sep 10 2018 12:41:49

%S 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,

%T 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,

%U 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

%N 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.

%H Ramanujan's Papers, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a> Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).

%F 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.

%e 0.4603233722587214303937620863844189747632149035387392240584250348445902629324...

%t 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

%Y Cf. A061502, A092742, A245074, A319090.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Sep 10 2018