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%I #37 Jan 15 2021 21:18:50
%S 0,0,0,0,0,4,4,3,8,2,6,9,3,1,2,5,0,6,9,5,3
%N Decimal expansion of imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).
%C For the decimal expansion of the real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function see A337404.
%C Sum_{m>=1} 1/(1/2 + i*z(m))^1 = 0.01154785448306... - i*A where 0.01154785448306 = A074760/2 and A > 10.5.
%C Sum_{m>=1} 1/(1/2 + i*z(m))^2 = -0.0230771586479... - i*0.000728434... where -0.0230771586479 = A245275/2
%C Sum_{m>=1} 1/(1/2 + i*z(m))^3 = -0.000055579115726... + i*0.0007262105... where -0.000055579115726 = A245276/2
%C Sum_{m>=1} 1/(1/2 + i*z(m))^4 = 0.0000368136106308... + i*0.0000044382...
%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/z(m)^6 = 0.0000001441739314...; see A335826.
%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 See A332645.
%F No explicit formula is known.
%e 0.000004438269312506953
%t (* 7-day-long procedure *)
%t kk = 0; Do[kk = kk + 1/(N[ZetaZero[n], 100])^4 , {n, 1, 1000000}]; Take[Join[{0, 0, 0, 0, 0}, RealDigits[Im[kk]][[1]]], 11]
%Y Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A335826.
%K nonn,cons,more
%O 0,6
%A _Artur Jasinski_, Aug 26 2020
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