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Search: id:A124288
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| A124288 |
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Indices of unstable zeros of the Riemann zeta function. |
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3
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| 1, 3, 6, 9, 13, 17, 21, 26, 30, 33, 40, 44, 50, 54, 61, 67, 70, 78, 79, 90, 93, 101, 109, 112, 117, 124, 134, 139, 147, 149, 153, 165, 167, 175, 186, 189, 197, 201, 214, 218, 219, 234, 235, 240, 253, 255, 266, 270, 275, 282, 288, 299, 300, 313, 317, 334, 342, 344
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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
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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. (to appear).
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
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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
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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
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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. 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.
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CROSSREFS
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Cf. A002410, A124289.
Sequence in context: A006590 A061781 A123753 this_sequence A002815 A109512 A025205
Adjacent sequences: A124285 A124286 A124287 this_sequence A124289 A124290 A124291
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KEYWORD
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hard,nonn
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AUTHOR
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Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Oct 24 2006, corrected Oct 29 2006
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EXTENSIONS
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Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 01 2006
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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