%I #34 Jan 15 2021 21:20:55

%S 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,

%T 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,

%U 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

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

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

%F 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).

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

%F Equals 1/4 - A338858.

%F Equals Sum_{k>=2} 1/(k*(k^4 - 1)). - _Vaclav Kotesovec_, Dec 24 2020

%e 0.0390670072379950810608...

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

%o (PARI) suminf(k=1, zeta(4*k+1)-1) \\ _Michel Marcus_, Dec 24 2020

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

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

%K nonn,cons

%O 0,2

%A _Artur Jasinski_, Dec 24 2020