Search: id:A124289
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%I A124289
%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 A. Fujii, Zeta zeros, Hurwitz zeta functions and L(1,Chi), Proc. Japan 
               Acad. 65 (1989), 139-142.
%D A124289 R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz 
               zeta-function, Math. Comp. (to appear).
%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 R. Garunkstis and J. Steuding, <a href="http://www.ams.org/mcom/0000-000-00/
               S0025-5718-06-01882-5/home.html">On the distribution of zeros of 
               the Hurwitz zeta-function</a>
%H A124289 J. Sondow and Eric Weisstein's World of Mathematics, <a href="http://
               mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</
               a>
%H A124289 M. Trott, <a href="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
%e A124289 Re(s) = 1/2 connect by paths of zeros of zeta(s,a) to zeros of zeta(s,
               1/2)
%e A124289 on the line Re(s) = 0, so rho78 and rho79 are "unstable twins," and 78 
               and 79 are members.
%Y A124289 Cf. A002410, A124288.
%Y A124289 Sequence in context: A098024 A117330 A033398 this_sequence A053083 A039435 
               A043258
%Y A124289 Adjacent sequences: A124286 A124287 A124288 this_sequence A124290 A124291 
               A124292
%K A124289 hard,nonn,more
%O A124289 1,1
%A A124289 Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Oct 24 2006
%E A124289 Corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Nov 10 
               2006, using more accurate calculations by R. Garunkstis and J. Steuding.

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