Re: zeta() extremly high up the critical strip

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

On Mon, Apr 20, 2026 at 07:45:33PM +0200, Bill Allombert wrote:
> On Mon, Apr 20, 2026 at 04:56:40PM +0200, Jeppe Stig Salling Nielsen wrote:
> > Hi all,
> > 
> > Seeing the page:
> > 
> > https://math.stackexchange.com/questions/2037020/
> > 
> > made me try to see if PARI/GP could handle it. I tried both with a floating-point input:
> > 
> > zeta(1.0/3 + I*5466322356764788987534453212467843688237746873357395.635356798779776664433)
> > 
> > and with an "exact" input:
> > 
> > zeta(1/3 + I*5466322356764788987534453212467843688237746873357395635356798779776664433/10^21)
> > 
> > and I tried with raising the `realprecison` default and the stack size.
> > 
> > On the stable version [2, 17, 3], I got "zeta: precision too low in mpcosm1.". I speculated this could be related to a previous thread on this list "Question on finding a Riemann Zeta function zero for high values of s", so I tried on the dev version [2, 18, 0]. But here I am getting "zeta: overflow in expo().".
> > 
> > I am not sure if it is reasonable to expect that PARI can calculate such a value, when the imaginary part of the input is almost 10^52?
> 
> For zeta(1/2+I*t) you should be able to reach t = 10^22
> 
> For zeta(2/3+I*t) you should be able to reach t = 10^22 too.
> 
> For zeta(1/3+I*t) you can only reach t=10^18 currently.
> (it uses the functional equation and choke on gamma(1/3+I*t))
> This is probably a bug, thanks for telling us about it!

This bug was fixed in the master branch.

On a 64 cores system:

? zeta(1/3+I*10^19)
%1 = 3807.6532704272757146592193505579849691 + 2154.9628063286881472300774220252755172*I
  ***   last result: cpu time 17min, 13,494 ms, real time 17,285 ms.
? zeta(1/3+I*10^20)
%3 = 175.40858272750429338016143524385898630 + 1845.4896537084166531641429926161157132*I
  ***   last result: cpu time 53min, 102 ms, real time 51,671 ms.
? zeta(1/3+I*10^21)
%4 = 511.88983503507988731887484616496413457 - 1928.6119704341230590475539996223486569*I
  ***   last result: cpu time 2h, 47min, 42,357 ms, real time 2min, 39,951 ms.

Thanks for reporting it!
Bill.