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A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

%I A335826 #11 Aug 06 2024 06:38:32
%S A335826 0,0,0,0,0,0,1,4,4,1,7,3,9,3,1,4,0,0,9,7,3,2,7,9,6,9,5,3,8,1,5,5,6,0,
%T A335826 9,4,8,2,0,9,0,7,0,3,6,8,8,3,0,0,8,5,0,9,0,9,8,1,1,8,7,1,5,9,9,9,3,6,
%U A335826 4,2,1,7,9,0,5,3,9,4,6,3,1,6,8,9,6,4,0,8,1,9,5,5,0,6,7,4,2,0,4,6,8,3,8,8,8,3,4,2,3,0,5
%N A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.
%C A335826 Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
%C A335826 Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
%C A335826 Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.
%C A335826 Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
%C A335826 Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
%C A335826 Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
%C A335826 Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
%C A335826 Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.
%H A335826 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 A335826 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 A335826 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 A335826 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 A335826 Universal formula for Sum_{n>=1} 1/z(n)^(2m) published in Voros 2002-2003 p. 22 (see Mathematica procedure below).
%e A335826 0.000000144173931400973279695381556....
%t A335826 m = 3; Join[{0, 0, 0, 0, 0, 0},RealDigits[N[((-1)^m (2^(2 m) - ((2^(2 m) - 1) Zeta[2 m] + (Zeta[2 m, 1/4] - Zeta[2 m, 3/4])/2^(2 m))/4 - (D[Log[Zeta[x]], {x, 2 m}] /. x -> 1/2)/(2 (2 m - 1)!) )), 105]][[1]]]
%Y A335826 Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815.
%K A335826 nonn,cons
%O A335826 0,8
%A A335826 _Artur Jasinski_, Jun 25 2020