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.
