A183030 Decimal expansion of Sum_{j>=1} tau(j)/j^3 = zeta(3)^2.

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

Offset: 1

Author

R. J. Mathar, Dec 18 2010

Keywords

• nonn
• cons

Comments

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.

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.

Examples

			1.4449407984336342339136.. = 1 +2/2^3 +2/3^3 +3/4^3 +2/5^3 +4/6^3 +2/7^3+...
		

Crossrefs

Cf. A000005, A002117, A098198, A175639, A183031.

Programs

Maple

evalf(Zeta(3)^2);

Mathematica

RealDigits[Zeta[3]^2, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PARI

zeta(3)^2 \\ Charles R Greathouse IV, Mar 04 2015

Formula

Equals the Euler product Product_{p= A000040} (1+ (2*p^s-1)/(p^s-1)^2) at s=3, or the square of A002117.