A339604 Decimal expansion of Sum_{k>=1} (zeta(3*k)-1).

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

Offset: 0

Author

Artur Jasinski, Dec 09 2020

Keywords

• nonn
• cons

Comments

For additional comments and generalization see attached text file.

Examples

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Links

• Artur Jasinski, Sums of zeta(p*n+q)-1
• T. J. Stieltjes, Table des valeurs des sommes Sk, Acta Math., Volume 10 (1887), 299-302.

Crossrefs

Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339605, A339606.

Programs

Mathematica

RealDigits[Chop[N[Sum[Zeta[3 n] - 1, {n, 1, Infinity}], 105]]][[1]]

PARI

suminf(k=1, zeta(3*k)-1) \\ Michel Marcus, Dec 09 2020

Formula

Equals Sum_{k>=2} 1/(k^3-1).

Equals 1 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).

Equals 1 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 - sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.

Equals 7/6 - Pi*tanh(Pi*sqrt(3)/2)/(2*sqrt(3)) - A339605/2.

Equals 4/3 - Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) + A339606.

Equals 1 - A339605 - A339606.