A339097 Decimal expansion of Sum_{k>=1} zeta(4*k+1)-1.

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

Offset: 0

Author

Artur Jasinski, Dec 24 2020

Keywords

• nonn
• cons

Examples

			0.0390670072379950810608...
		

Crossrefs

Cf. A256919 (4*k), A339083 (4*k+2), A338858 (4k+3).

Cf. also A338815, A339135, A339529, A339530, A339604, A339605, A339606.

Programs

Mathematica

Join[{0},RealDigits[N[Re[Sum[Zeta[4 n + 1] - 1, {n, 1, Infinity}]], 105]][[1]]]

PARI

suminf(k=1, zeta(4*k+1)-1) \\ Michel Marcus, Dec 24 2020

Formula

Equals Sum_{k>=2} (k^3 -3*k^2 + k - 2)/(k^5 - k).

Equals 3/8 - gamma/2 - Re(Psi(i))/2, where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).

Equals 3/8 - Re(H(I))/2, where H is the harmonic number function.

Equals 1/4 - A338858.

Equals Sum_{k>=2} 1/(k*(k^4 - 1)). - Vaclav Kotesovec, Dec 24 2020