NewtonZero never comes down

• To: mathgroup at smc.vnet.net
• Subject: [mg51416] NewtonZero never comes down
• From: Roger Bagula <tftn at earthlink.net>
• Date: Sat, 16 Oct 2004 04:21:07 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```I've tried several ways to find better values for
zeta zeros fast and easy...
FindRoot[] just doesn't give much accuracy.
NewtonZero[] tends to just spin it's wheels on my machine....
Here's what I came up with that gets about 20 places
and pretty fast relatively.
It's much better than FindRoot[] anyway!
The zeta zeros all seem to be transcendental irrational numbers
and are harder to calculate than most numbers in modern number theory.

(* finding Zeta Zero roots*)
(* using approximation from analyitic continuation on 0<=z<=1 strip
where zeta dosesn't converge*)
Clear[RSzeta,t,g3,f,g,a]
RSzeta[t_]=RiemannSiegelZ[t]/Exp[I*RiemannSiegelTheta[t]]
Digits=10000
t=(2 Sqrt[2])*25*Pi^2*Sqrt[n]/(6*Sqrt[Digits])-16
(* gives the zeta zeros given in Jahnke-Emde Tables of Functions, Dover*)
g3=Delete[Union[Table[If[N[Abs[RSzeta[t]]]<0.01,N[t,20],0],{n,1,Digits}]],1]
Dimensions[g3]
ListPlot[g3,PlotJoined->True]
(* definition of Newton function*)
f[x_]:=N[x-Zeta[x]/Zeta'[x],20]
(* ten iterations toward the zero*)
g[x_]:=NestList[f,1/2+I*x,10][[10]]
(* values starting at the continuation values approximated*)
a=Table[g[g3[[n]]],{n,1,Dimensions[g3][[1]]}]
ListPlot[Im[a],PlotJoined->True]

Respectfully, Roger L. Bagula