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%I A183030 #16 May 22 2023 02:33:26
%S A183030 1,4,4,4,9,4,0,7,9,8,4,3,3,6,3,4,2,3,3,9,1,3,6,8,5,0,7,8,8,0,6,9,8,7,
%T A183030 8,2,7,1,8,3,7,3,5,4,0,5,7,6,3,8,8,8,6,7,4,1,3,1,4,3,4,1,6,1,8,9,8,5,
%U A183030 8,3,8,5,6,1,3,1,3,5,4,1,0,1,9,6,6,1,9
%N A183030 Decimal expansion of Sum_{j>=1} tau(j)/j^3 = zeta(3)^2.
%C A183030 This is the zeta-function Sum_{j>=1} A000005(j)/j^s evaluated at s=3. At s=2 we find A098198, at s=4 A183031.
%C A183030 Since tau(n)/n^3 is a multiplicative function, one finds an Euler product for the sum, which is expanded with an Euler transformation to a product of Riemann zeta functions as in A175639 for numerical evaluation.
%F A183030 Equals the Euler product Product_{p= A000040} (1+ (2*p^s-1)/(p^s-1)^2) at s=3, or the square of A002117.
%e A183030 1.4449407984336342339136.. = 1 +2/2^3 +2/3^3 +3/4^3 +2/5^3 +4/6^3 +2/7^3+...
%p A183030 evalf(Zeta(3)^2);
%t A183030 RealDigits[Zeta[3]^2, 10, 120][[1]] (* _Amiram Eldar_, May 22 2023 *)
%o A183030 (PARI) zeta(3)^2 \\ _Charles R Greathouse IV_, Mar 04 2015
%Y A183030 Cf. A000005, A002117, A098198, A175639, A183031.
%K A183030 nonn,cons
%O A183030 1,2
%A A183030 _R. J. Mathar_, Dec 18 2010

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