|
|
A124288
|
|
Indices of unstable zeros of the Riemann zeta function.
|
|
2
|
|
|
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, 355, 359, 370, 371, 384, 387, 394, 409, 418, 422, 431, 434, 444, 450, 459, 465, 477, 489, 493, 500, 501
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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.
|
|
REFERENCES
|
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".
|
|
LINKS
|
|
|
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).
|
|
EXAMPLE
|
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.
|
|
CROSSREFS
|
|
|
KEYWORD
|
hard,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Corrected by Jonathan Sondow, Nov 10 2006, using more accurate calculations by R. Garunkstis and J. Steuding.
|
|
STATUS
|
approved
|
|
|
|