Re: Question on finding a Riemann Zeta function zero for high values of s

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Thu, Oct 23, 2025 at 11:38:43AM +0200, Cohen Henri wrote:
> 
> Second attempt at a patch, the first was wrong. Replace the computation of p1 by
>  p1 = gsub(gmul(pmd, mulcxI(gsqr(zn))), gmul(s, glog(zn, prec)));
>  p1 = gmul(gexp(p1, prec), gmul(aleps, gadd(x, ix)));
>  
> This at least gives the right result for the zero at height 10^11, and does not crash at 10^21.

Your patch gives this

? s = 1/2 +  1370919909931995308226.68016095*I
%1 = 1/2+1370919909931995308226.6801609500000000*I
? zeta(s)
%2 = 4.0536436970172441408698440140242774312E-8+2.4195172994094734771061685252890615819E-8*I
? ##
  ***   last result: cpu time 3h, 5min, 32,685 ms, real time 9min, 25,142 ms.
(which is the same result as mine and looks OK given the input accuracy).

A better approximation of s is
? s2 = 1/2+1370919909931995308226.68016095354847361*I;
? zeta(s2)
%47 = -1.93955520903444813832202660734386359006E-15-1.15767118329892041096332490256606300204E-15*I

Cheers,
Bill