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A124289 Unstable twins = pairs of consecutive numbers in A124288 (indices of unstable zeros of the Riemann zeta function).

%I A124289 #23 Feb 16 2025 08:33:03
%S A124289 78,79,218,219,234,235,299,300,370,371,500,501
%N A124289 Unstable twins = pairs of consecutive numbers in A124288 (indices of unstable zeros of the Riemann zeta function).
%C A124289 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 A124289 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 A124289 M. Trott, The Mathematica GuideBook for Symbolics, Springer-Verlag, 2006, see "Zeros of the Hurwitz Zeta Function".
%H A124289 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 A124289 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 A124289 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 A124289 J. Sondow and Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%H A124289 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 A124289 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 A124289 The consecutive zeros rho78 and rho79 of zeta(s,1) on the line Re(s) = 1/2 connect by paths of zeros of zeta(s,a) to zeros of zeta(s,1/2) on the line Re(s) = 0, so rho78 and rho79 are "unstable twins," and 78 and 79 are members.
%Y A124289 Cf. A002410, A124288.
%K A124289 hard,nonn,more
%O A124289 1,1
%A A124289 _Jonathan Sondow_, Oct 24 2006
%E A124289 Corrected by _Jonathan Sondow_, Nov 10 2006, using more accurate calculations by R. Garunkstis and J. Steuding.