Re: Zeta zero information on LMFDB

[John Jones]

Hi,

Zeta zeros is not my particular expertise, so I am copying this to the LMFDB support group.  Perhaps someone there can respond.

John Jones

On Mon, Apr 13, 2026 at 6:26 PM Sean Ashenmil <ashenm...@gmail.com> wrote:

Hey john! I believe there is one missing piece of information ( one number/integer) that alot of people just throw the cold shoulder on but i personally believe, can tell us alot about the zeta function and RH operations, and it really is pretty simple because its already in the data. And this number is - the last zeta zero ( its integer, before its infinite decimal tail ) - before which every zero after this one contributes to a infinite consecutive integer cluster where there are no gaps inbetween the actual integers of the zeros, so the one im looking for- would have a integer gap right after, but it would be the last one, because although not proven yet, it sure looks like there will never ever be a gap in between integers after that last 103 billionth zero that  LMFDB holds.  So all one would have to do is design a program/algorithm from the last zero on down from the LMFDB database until it finds this gap - now im asking you this just in case you have access to this information to where you might not need to program something to search, but nontheless, do you think you can help with this at all, because i believe we can measure ALOT from this certain number,  but thanks for reading this!

[David W. Farmer]

I would characterize that idea as entertainment math, rather
than research.

It is clear that there is a point T, after which the consecutive
zeros of the zeta function have their imaginary parts separated
by less than 1. Equally clear, there is the actual value of T,
and there is a value which we can prove. And maybe we can do an
explicit calculation which shows that those two numbers are equal.

But there is no research question here: it is just a matter of
doing it, if someone feels like going to the effort. That work
does not involve new mathematics.

If I have missed that there is some hidden meaning to that number,
I am happy to listen. But I do not see that there is anything
special to the last point at which the zeta zeros are separated
by some specific number \delta. And \delta = 1 is not special.

Apologies if I have misunderstood the suggestion: please
enlighten me.

Regards,

David

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[David Lowry-Duda]

On 02:20 Tue 14 Apr 2026, David W. Farmer wrote:
>
>It is clear that there is a point T, after which the consecutive
>zeros of the zeta function have their imaginary parts separated
>by less than 1. Equally clear, there is the actual value of T,
>and there is a value which we can prove. And maybe we can do an
>explicit calculation which shows that those two numbers are equal.

Here is one route. In Titchmarsh (Theory of the Riemann Zeta Function), Theorem 9.12 shows that there exist A, T such that $\zeta(s)$ has a zero $\beta + i \gamma$ with $| \gamma - T | < A / (\log \log \log T) $, for all sufficiently large T.

The proof uses Borel-Caratheodory but in a non-effective way. It could be made effective, both in A and T. I expect that the resulting bound would be small enough to be contained within David Platt's computation and the LMFDB.

I don't think this would be hard to work out, but to my knowledge it hasn't been written down.

Of course, this is also much weaker than what we expect to be true.

--
David Lowry-Duda <da...@lowryduda.com> <davidlowryduda.com>

[David W. Farmer]

It is not a matter of can we do it. Clearly: yes.

It is a question of whether it is actually interesting
and worth putting on the LMFDB.

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[David Lowry-Duda]

On 04:27 Tue 14 Apr 2026, David W. Farmer wrote:
>
>It is not a matter of can we do it. Clearly: yes.
>
>It is a question of whether it is actually interesting
>and worth putting on the LMFDB.

You're right, I didn't comment on that.

I don't think this should be in the LMFDB. It would be one thing if this
quantity were arithmetically interesting. But in my opinion it is instead a cute
statistic. This can be fun, but I don't think it connects to other families of
L-functions for example.

- DLD