A124289 Unstable twins = pairs of consecutive numbers in A124288 (indices of unstable zeros of the Riemann zeta function).

78, 79, 218, 219, 234, 235, 299, 300, 370, 371, 500, 501

Offset: 1

Author

Jonathan Sondow, Oct 24 2006

Keywords

• hard
• nonn
• more

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 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.
		

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, A124288.

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