A335918 - OEIS

%I #12 Jul 30 2020 06:55:28

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

%T 7,9,7,1,3,5,9,6,2,9,1,9,7,9,9,5,6,5,2,8,7,0,1,9,3,5,6,9,0,9,1,7,9,0,

%U 0,0,3,6,7,0,3,7,8,2,2,0,4,4,7,1,4,6,4,8,7,5,7,0,0,6,2,8,5,8,5,8,4,5,5,0,0,5,8,4,8

%N Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2)^2 where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function.

%C Sum_{m>=1} 1/z(m) is a divergent series; see A332614.

%C Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.

%C Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.

%C Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.

%C Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.

%C Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.

%C Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.

%C Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.

%H André Voros, <a href="https://arxiv.org/abs/math/0104051">Zeta functions for the Riemann zeros</a>, arXiv:math/0104051 [math.CV], 2002-2003, p.22 Table 1.

%F Equals: 3 + gamma + gamma^2 - Pi^2/8 - log(4*Pi) + 2*gamma(1), where gamma is the Euler-Mascheroni gamma constant (see A001620) and gamma(1) is 1st Stieltjes constant (see A082633).

%e 0.0000371006364374648715125054339132797135962919799565287...

%t Join[{0, 0, 0, 0},RealDigits[N[3 + EulerGamma + EulerGamma^2 - Pi^2/8 - Log[4 Pi] + 2 StieltjesGamma[1], 105]][[1]]]

%Y Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815.

%K nonn,cons

%O 0,5

%A _Artur Jasinski_, Jun 29 2020