As part of an ongoing research project, and also to learn how to create animated graphs, I decided to perform a literature review of zero density estimates [N(\sigma,T) \ll T^{A(\sigma)(1-\sigma)+o(1)}] for the Riemann zeta function, where (1/2 < \sigma < 1) and the game is to get the exponent (A(\sigma)), and particularly the supremum (\sup_\sigma A(\sigma) ), as small as possible. The Riemann hypothesis basically asserts that this supremum is 0, while the weaker Density hypothesis asserts that this supremum is at most 2. By 1972, the work of Ingham and Huxley had pushed the supremum down to 12/5=2.4, but it then remained stuck for over fifty years, until the recent work of Guth and Maynard reduced this to 30/13=2.307...
I had always wondered why there did not seem to be a comprehensive survey of all the zero density theorems that had been established over the years, and now I know why: the literature is immensely complicated, especially in the region (3/4 \leq \sigma < 1) where there has been a lot of activity using a variety of methods. The bounds tend be piecewise in nature, mostly due to the fact that the methods rely on controlling integer moments rather than fractional moments. However, while these bounds are quite messy to state in human-readable form, they are quite digestible to a computer, and it was surprisingly routine to collate all the bounds into a single Python file, which I then used to create the attached animation. These zero density estimates are useful inputs to other analytic number theory problems, so when our project concludes we will be able to easily tweak the code to explore what could have been proven at different points in history of the subject.
Small update: I did find a recent table of zero density estimates at https://arxiv.org/pdf/2306.05599 which also contained some estimates that I was not aware of, so I plan to merge them into a single comprehensive review of the subject.
As always, you're doing a great job of making sense of what's out there.
However, it seems to me that, in general, prime number problems, especially the major ones, have been shrouded in incidental complexity due to research. In other words, I feel that a better job should be done to determine simpler creative solutions. Classics. Not as long as Wiles proof!
This has been my goal over the past few years of working on such concepts. Cheers.
@tao Cool, nice to see. Would it be possible to get a static image of the final frame? I wanted to eyeball that final bounding function, but of course it went away and waiting for the gif to run its course is rather slow...
Needless to say, I did use some AI assistance to help write the code. To some extent, I view the AI assistance as psychological support - it is much easier to convince myself to spend time coding when I know that, in addition to my older methods of debugging and googling, I also have the AI tools to provide initial code, and autocomplete partially written code, though I did find that they were not particularly useful at debugging. Still, GPT was able to quickly create a simple animation of some test functions that compiled on the first try; it still took me the better part of an hour to tweak and debug it to the actual animnation I wanted, though.
Here I am hiding almost all of my math (philosophy, inventions, etc.) work over the past 20 years... And you're putting the stuff into GPT systems. Whoa.
I have no clue how you have come to have as much trust as you do in this stuff. I barely even share my ideas anymore. Only pen and paper know them.
Either way, I'm excited to see you posting here, man.
Back to weird densities again today. Terrible moment equations so far. Ugh. No GPT.
@oblomov Sure. Unconditionally, we have the red curve in this image. Assuming the density conjecture, we can lower to the green curve. Assuming the Lindelof hypothesis, we can drop the exponent all the way to zero past 3/4. Assuming Riemann, the entire curve drops to zero (except at sigma=1/2, of course).