A124288 Indices of unstable zeros of the Riemann zeta function.

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

Offset: 1

Author

Jonathan Sondow, Oct 24 2006, corrected Oct 29 2006

Keywords

• hard
• nonn

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.

Examples

			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.
		

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

• 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. Comput. 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.
• 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

Crossrefs

Cf. A002410, A124289.

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).

Extensions

Corrected by T. D. Noe, Nov 01 2006

Corrected by Jonathan Sondow, Nov 10 2006, using more accurate calculations by R. Garunkstis and J. Steuding.