%I #20 Aug 06 2024 06:37:57
%S 0,0,0,0,3,7,1,7,2,5,9,9,2,8,5,2,6,9,6,8,6,1,6,4,8,6,6,2,6,2,4,7,1,7,
%T 4,0,5,7,8,4,5,3,6,5,0,8,8,9,7,3,0,0,8,3,2,1,3,5,7,5,5,0,6,3,7,1,8,4,
%U 6,1,3,3,2,9,8,8,4,5,7,2,8,1,3,7,2,9,7,6,0,3,5,7,2,3,3,7,4,2,4,2,9,6,0,2,8,3,7,0,0
%N Decimal expansion of Sum_{n>=1} 1/z(n)^4 where z(n) is the imaginary part of the n-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...; this sequence.
%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.25 Table 2.
%H André Voros, <a href="https://citeseerx.ist.psu.edu/pdf/937e9d9c737304007c6af6a58901a8c1e530df31">Zeta functions for the Riemann zeros</a>, 2001(2008) p.20 Table 1.
%H André Voros, <a href="https://doi.org/10.5802/aif.1955">Zeta functions for the Riemann zeros</a>, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
%H André Voros, <a href="https://doi.org/10.1007/978-3-642-05203-3">Zeta functions over Zeros of the Zeta functions</a>, 2010, p. 153.
%F Equals 16-Pi^4/24+(Zeta[4,3/4]-Zeta[4,1/4])/64-(Log[Zeta[x]]''''[1/2])/24
%e 0.0000371725992852696861648662624717405784536508897300...
%t Join[{0,0,0,0},RealDigits[N[-1/12*(D[Log[Zeta[x]],{x,4}]/. x -> 1/2) - 1/24 Pi^4 -(Zeta[4, 1/4] - Zeta[4, 3/4])/64 + 16, 105]][[1]]]
%Y Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814.
%K nonn,cons
%O 0,5
%A _Artur Jasinski_, Jun 25 2020