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

1,1

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

Table of n, a(n) for n=1..12.

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

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 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.

Crossrefs

Cf. A002410, A124288.

Sequence in context: A033398 A204376 A176094 * A181467 A344812 A345476

Adjacent sequences: A124286 A124287 A124288 * A124290 A124291 A124292

Keyword

hard,nonn,more

Author

Jonathan Sondow, Oct 24 2006

Extensions

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

Status

approved