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@tao How do the coefficients appear? I would appreciate anyone help just to not waste professor Tao's time if possible, just read the Scientific American article and while this is beyond me also watched both videos and gave a look to the paper but I'm still feel confused, in the first video professor Maynard says that if $N(\sigma ,T)\ll T^{A(1-\sigma )+o(1)}$ then the asymptotics for primes in short intervals are $x^{(1-1/A)}$, and that Ingham and Huxley gave the bounds $N(\sigma ,T)\le T^{\frac{3(1-\sigma )}{2-\sigma }+o(1)}$ and $N(\sigma ,T)\le T^{\frac{3(1-\sigma )}{3\sigma -1}+o(1)}$ respectively and in the new paper they introduce the bound $N(\sigma ,T)\le T^{\frac{15(1-\sigma )}{3+5\sigma }+o(1)}$, so the Ingham and Huxley bounds have the same exponent for $\sigma =3/4$ and the Ingham and Guth-Maynard bounds have the same exponent for $\sigma =7/10$ and so plugging those values in their respective denominators gives $T^{\frac{12}{5}(1-\sigma )+o(1)}$ and $T^{\frac{30}{13}(1-\sigma )+o(1)}$, therefore they say that the exponent improved from A=12/5 to A=30/13 (theorem 1.2) or equivalently from 1-1/A=7/12 to 1-1/A=17/30 (Corollary 1.3) but I'm confused because these are taken around different points.

They also say that the Ingham and Huxley estimate around $\sigma =3/4$ bound the error R of $N(\sigma ,T)$ by $R\le T^{\frac{3}{5}+o(1)}$ and that they improve that to $R\le T^{\frac{13}{25}+o(1)}$ (theorem 1.1) but how is that equivalently to improve the asymptotics from 1/6 to 2/15, I mean there is no clear translation as with the pair (A, 1-1/A) from before.

@dabed Here is a plot of the key upper bounds on the exponent $A(\sigma )$ in the zero density estimate $N(\sigma ,T)\le T^{A(\sigma )(1-\sigma )+o(1)}$. For applications to the primes, the key quantity is 𝐴, which is the supremum of all the $A(\sigma )$. Prior to the work of Guth and Maynard, the supremum was attained at $\sigma =3/4$; but as one can see from the graph, the new bound of Guth and Maynard has shifted the location where the supremum is attained to $\sigma =7/10$.

@tao Ah I see, sorry for making you draw it for me and sorry again as it was clear too in the previous graph that you posted, I should have realized this even without graphs and yet here I am.

About the improvement of the asymptotics from 1/6 to 2/15, I did a search ctrl+f+1/25 and got a second hit in the last page (p. 48) and it seems these numbers appear after some math that it isn't done in the paper and which I probably wouldn't understand anyway as I read only until section 2 (p. 8) grasping a mere fraction of it.

May I bother you a bit more to ask you what prompt you used for the first graph or some tips to reproduce it? I tried a bit but didn't get even a little close

@tao Thanks! I created a subsection about the density hypothesis on the wikipedia article on the Lindelöf hypothesis with what I was able to grasp. (ttps://en.wikipedia.org/wiki/Lindelöf_hypothesis#The_density_hypothesis)

I can't use a third party image that has no Creative Commons license, that is why I needed the code to create my own version and upload it with CC license, in the Wikimedia summary I clarified it was generated using your code, and the caption is also the same you wrote.

If the Riemann hypothesis would push the diagram down to the x-axis then the RH is kind of equivalent to proving $A(\sigma )$=0 or did I understand wrong?

@dabed The RH implies that $A(\sigma )=0$ for $\sigma >1/2$, but is even stronger than that claim. For instance, if there were a finite number of zeroes of zeta to the right of the critical strip, then $A(\sigma )$ would still be zero for $\sigma >1/2$, but the Riemann hypothesis would now be false.

There are also several more known zero density theorems beyond the ones of Ingham, Huxley, and Guth-Maynard, but the complete picture is rather complicated to state, and these theorems only give improvements in the range $\sigma >3/4$ (this is why Guth-Maynard was a significant breakthrough, as it was the first in decades to affect the critical value $\sigma =3/4$. See Table 2 of https://arxiv.org/abs/2306.05599 .

@tao Ah right thanks I was clearly playing with the quantities without thinking correctly about their meaning.

While I don't want to trouble you asking more, truth is I don't understand enough to generate anymore questions.

I just saw now that ArXiv also contains a blog post of yours where the paper is cited and that you were working on the same even before being aware of it, funny to see how ideas converge specially when evidently it isn't an easy task with the vastness of all the previous results.

Thanks a lot for your help and all the best!