A381028 Decimal expansion of Sum_{k>=1} zeta(2k)/((2k-1)*2^(2k)).

4, 3, 7, 6, 5, 8, 2, 4, 2, 3, 1, 1, 2, 6, 1, 0, 9, 3, 3, 1, 5, 9, 2, 0, 9, 2, 6, 4, 3, 8, 0, 5, 1, 4, 0, 1, 6, 4, 8, 4, 3, 5, 6, 4, 5, 3, 5, 2, 3, 0, 6, 9, 6, 8, 3, 0, 2, 7, 1, 5, 6, 1, 3, 1, 5, 1, 3, 3, 2, 3, 4, 3, 5, 7, 1, 5, 8, 9, 4, 1, 7, 2, 4, 1, 6, 0, 1, 6, 8, 3, 9, 4, 9, 8, 3, 0, 9, 8, 5, 4, 2, 3, 9, 3, 1

Offset

0,1

Comments

Including the term k=0 with zeta(0) = -1/2 gives 0.937658... = this + 1/2.

Links

Table of n, a(n) for n=0..104.

H. M. Srivastava, M. L. Glasser, and V. S. Adamchik, Some definite integrals associated with the Riemann Zeta Function, Z. Anal. Anw. 19 (3) (2000) 831-846, (2.18) at n=0.

Formula

4*this = 1.75063296924504437... = Sum_{k>=1} (1/k)*log((2k+1)/(2k-1)).

Equals Sum_{k>=1} arctanh(1/(2*k))/(2*k) = Sum_{k>=1} arccoth(2*k)/(2*k). - Amiram Eldar, Feb 15 2025

Example

0.4376582423112610933159209...

Maple

evalf(Sum(Zeta(2*k)/((2*k-1)*4^k), k = 1 .. infinity), 105) # Amiram Eldar, Feb 15 2025

Mathematica

RealDigits[NSum[Zeta[2*k]/((2*k - 1)*4^k), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 200]][[1, 1 ;; 105]] (* Amiram Eldar, Feb 15 2025 *)

Prog

(PARI) sumpos(k=1, zeta(2*k)/((2*k-1)*2^(2*k))) \\ Michel Marcus, Feb 13 2025

Crossrefs

Cf. A256318, A355922.

Sequence in context: A388465 A046560 A205392 * A131413 A249189 A168200

Adjacent sequences: A381025 A381026 A381027 * A381029 A381030 A381031

Keyword

nonn,cons

Author

R. J. Mathar, Feb 12 2025

Extensions

More terms from Amiram Eldar, Feb 15 2025

Status

approved