A183031 - OEIS

A183031

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

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, 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, 4, 6, 7, 7, 8, 0, 0, 3, 8, 7, 3, 6, 1, 6, 7, 9, 4

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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.

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.

LINKS

Table of n, a(n) for n=1..85.

Index entries for transcendental numbers

FORMULA

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.

EXAMPLE

1.1714235822309350626084... = 1 + 2/2^4 + 2/3^4 + 3/4^4 + 2/5^4 + 4/6^4 + 2/7^4 + ...

MAPLE

evalf(Pi^8/8100) ;

MATHEMATICA

RealDigits[Zeta[4]^2, 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PROG