Re: zeta() extremly high up the critical strip

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

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!

Cheers,
Bill