Re: possible problem with Lehmer pairs for the Riemann zeta function values

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

On Thu, Oct 23, 2025 at 08:54:04PM -0700, American Citizen wrote:
> To all:
> 
> I was able to successfully set up Newton's method using complex variables
> and the zeta function so I could find the best s value for a particular
> given s value for a Riemann zeta function zero.
> 
> In examining a particular Lehmer pair, the GP Pari calculated values do NOT
> match the given values
> 
> > REF: https://www.lmfdb.org/zeros/zeta/?limit=4&N=1048449113

After some research, I think it is due to conversion to base 10 in the original
data.

Let us convert both result to base 2;

? dbg_x(388858886.0022851217767970582610330824019)
[&=00007f3c4c9c2e68] REAL(lg=5):0400000000000005 (+,expo=28):600000000000001c b96c203004ae0fda 63c7b7fa00102bc3 a712acfdfa660a1a
? dbg_x(388858886.0022851217767970582610330824021)
[&=00007f3c4c9c2e68] REAL(lg=5):0400000000000005 (+,expo=28):600000000000001c b96c203004ae0fda 63c7b7fa00102bc3 c78659c90d370957

We see that they are actually identical up to 128bit (and even 129, since a = 1010 and c = 1100).

What is likely is that the original result was computed with correct rounding in base 2
with 128 bit of accuracy, and then converted to base 10. Unfortunately it is
not possible to get correct rounding in base 10 this way in all cases.

In fact I believe the result was displayed with 40 digits of accuracy, even
while knowing the last two could be incorrect, because this allows to recover
the correct 128bit value by converting back to base 2.
If the result was displayed with only 38 digits, it would not have been
possible to get back the correct 128bit by conversion.

Cheers,
Bill.