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A124289
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Unstable twins = pairs of consecutive numbers in A124288 (indices of unstable zeros of the Riemann zeta function).
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2
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78, 79, 218, 219, 234, 235, 299, 300, 370, 371, 500, 501
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
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REFERENCES
| A. Fujii, Zeta zeros, Hurwitz zeta functions and L(1,Chi), Proc. Japan Acad. 65 (1989), 139-142.
R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz zeta-function, Math. Comp. 76 (2007), 323-337.
R. Garunkstis and J. Steuding, Questions around the Nontrivial Zeros of the Riemann Zeta-Function. Computations and Classifications, Math. Model. Anal. 16 (2011), 72-81.
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.
M. Trott, The Mathematica GuideBook for Symbolics, Springer-Verlag, 2006, see "Zeros of the Hurwitz Zeta Function".
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LINKS
| R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz zeta-function
J. Sondow and Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
M. Trott, Zeros of the Generalized Riemann Zeta Function zeta(s,a) as a Function of a
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FORMULA
| 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).
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EXAMPLE
| 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.
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CROSSREFS
| Cf. A002410, A124288.
Sequence in context: A033398 A204376 A176094 * A181467 A053083 A039435
Adjacent sequences: A124286 A124287 A124288 * A124290 A124291 A124292
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KEYWORD
| hard,nonn,more
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AUTHOR
| Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Oct 24 2006
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EXTENSIONS
| 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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