%I #20 Mar 18 2019 09:36:11
%S 1,3,6,9,13,17,21,26,30,33,40,44,50,54,61,67,70,78,79,90,93,101,109,
%T 112,117,124,134,139,147,149,153,165,167,175,186,189,197,201,214,218,
%U 219,234,235,240,253,255,266,270,275,282,288,299,300,313,317,334,342,344,355,359,370,371,384,387,394,409,418,422,431,434,444,450,459,465,477,489,493,500,501
%N Indices of unstable zeros of the Riemann zeta function.
%C Assuming the Riemann Hypothesis, the nonreal zeros of zeta(s,1) = zeta(s) lie on the critical line Re(s) = 1/2 and the nonreal zeros of zeta(s,1/2) = (2^s - 1)*zeta(s) lie on the critical line and on the imaginary axis Re(s) = 0.
%D M. Trott, Zeros of the Generalized Riemann Zeta Function zeta(s,a) as a Function of a, background image in graphics gallery, in S. Wolfram, The Mathematica Book, 4th ed. Cambridge, England: Cambridge University Press, 1999, p. 982.
%D M. Trott, The Mathematica GuideBook for Symbolics, Springer-Verlag, 2006, see "Zeros of the Hurwitz Zeta Function".
%H A. Fujii, <a href="https://doi.org/10.3792/pjaa.65.139">Zeta zeros, Hurwitz zeta functions and L(1,Chi)</a>, Proc. Japan Acad. 65 (1989), 139-142.
%H R. Garunkstis and J. Steuding, <a href="https://doi.org/10.1090/S0025-5718-06-01882-5">On the distribution of zeros of the Hurwitz zeta-function</a>, Math. Comput. 76 (2007), 323-337.
%H R. Garunkstis and J. Steuding, <a href="https://klevas.mif.vu.lt/~garunkstis/preprintai/classzerosMMA.pdf">Questions around the Nontrivial Zeros of the Riemann Zeta-Function. Computations and Classifications</a>, Math. Model. Anal. 16 (2011), 72-81.
%H J. Sondow and Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%H M. Trott, <a href="https://web.archive.org/web/20071210074226/http://documents.wolfram.com/v4/MainBook/G.2.22.html">Zeros of the Generalized Riemann Zeta Function zeta(s,a) as a Function of a</a>
%F Solve the differential equation ds(a)/da = -(dzeta(s,a)/da)/(dzeta(s,a)/ds) = s*zeta(s+1,a)/(dzeta(s,a)/ds) where s = s0(a) and zeta(s0(a),a) = 0. For initial conditions use the zeros of zeta(s,1).
%e The first zero rho1 of zeta(s,1) on the line Re(s) = 1/2 connects by a path of zeros of zeta(s,a) to a zero of zeta(s,1/2) on the line Re(s) = 0, so rho1 is "unstable" and 1 is a member.
%e The 2nd zero rho2 of zeta(s,1) on Re(s) = 1/2 connects to a zero of zeta(s,1/2) on Re(s) = 1/2, so rho2 is "stable" and 2 is not a member.
%Y Cf. A002410, A124289.
%K hard,nonn
%O 1,2
%A _Jonathan Sondow_, Oct 24 2006, corrected Oct 29 2006
%E Corrected by _T. D. Noe_, Nov 01 2006
%E Corrected by _Jonathan Sondow_, Nov 10 2006, using more accurate calculations by R. Garunkstis and J. Steuding.