cp's OEIS Frontend

A183031 Decimal expansion of Sum_{j>=1} tau(j)/j^4 = Pi^8/8100.

%I A183031 #18 May 22 2023 02:32:53
%S A183031 1,1,7,1,4,2,3,5,8,2,2,3,0,9,3,5,0,6,2,6,0,8,4,6,6,1,1,1,5,9,3,4,2,7,
%T A183031 8,7,6,1,3,5,4,5,4,2,5,5,7,5,8,1,5,8,3,5,7,0,5,0,6,2,8,5,6,9,7,6,1,3,
%U A183031 4,6,7,7,8,0,0,3,8,7,3,6,1,6,7,9,4
%N A183031 Decimal expansion of Sum_{j>=1} tau(j)/j^4 = Pi^8/8100.
%C A183031 This is the zeta-function Sum_{j>=1} A000005(j)/j^s evaluated at s=4. At s=2, we find A098198; at s=3, A183030.
%C A183031 Since tau(n)/n^4 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.
%H A183031 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A183031 Equals the Euler product Product_{p prime} (1 + (2*p^s - 1)/(p^s - 1)^2) at s=4, which is the square of A013662.
%e A183031 1.1714235822309350626084... = 1 + 2/2^4 + 2/3^4 + 3/4^4 + 2/5^4 + 4/6^4 + 2/7^4 + ...
%p A183031 evalf(Pi^8/8100) ;
%t A183031 RealDigits[Zeta[4]^2, 10, 120][[1]] (* _Amiram Eldar_, May 22 2023 *)
%o A183031 (PARI) zeta(4)^2 \\ _Charles R Greathouse IV_, Mar 04 2015
%Y A183031 Cf. A000005, A098198, A183030, A175639.
%K A183031 nonn,cons
%O A183031 1,3
%A A183031 _R. J. Mathar_, Dec 18 2010