1,1

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.

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".

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

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).

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.

Cf. A002410, A124288.

Sequence in context: A098024 A117330 A033398 this_sequence A053083 A039435 A043258

Adjacent sequences: A124286 A124287 A124288 this_sequence A124290 A124291 A124292

hard,nonn,more

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Oct 24 2006

Corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Nov 10 2006, using more accurate calculations by R. Garunkstis and J. Steuding.