Contradictory implicit assumptions in auto_cheby

Stream: PrimeNumberTheorem+

[Harald Helfgott]

I suspect this matter may have already been spotted and I can't just find it. There are issues with the proposed proof of auto_cheby (a crucial step in the second version of the Tauberian argument - its formalization is still incomplete).

First of all, there are some obvious typos in the blueprint - for one thing, $\int_{\mathbb{R}}\hat{\psi}\left(\frac{1}{2\pi} u\right) e^u du$ should clearly be $n \int_{\mathbb{R}}\hat{\psi}\left(\frac{1}{2\pi} u\right) e^{-u} du$. At any rate, that is immaterial for what follows.

What is more serious is that you are requiring $\hat{\psi}$ to have one-sided exponential decay. That's incompatible with $\psi$ being compactly supported. (This is perhaps not as well-known and easy to prove as the statement that $\hat{\psi}$ having two-sided exponential decay is incompatible with $\psi$ being compactly supported, but I think it's still standard.)

[Terence Tao]

Technically, the proof still works even if this integral is infinite (one gets a better bound in which there is a weight that blows up somewhat as n is small), but the proof becomes conceptually a bit weird. I've just uploaded at https://github.com/AlexKontorovich/PrimeNumberTheoremAnd/pull/250 a revised version in which the previous corollary gives this bound on a dyadic type interval $[(1-\varepsilon) x, x]$, and then one performs a strong induction on $x$ to conclude. (I was hoping to use an integration trick and Fubini to avoid induction (or summing a geometric series); the argument still works, but looks suspicious since it now involves divergent integrals.)

[Harald Helfgott]

Yes, that should do it.

Incidentally, there is also the alternative route of giving a Wiener-Ikehara proof $M(x) = o(x)$ (avoiding this issue altogether, since $\sum_{n\leq x} |\mu(n)|\leq x$ is obvious) - that requires a bit of extra work in some ways but also does away with some other work, I'd think.

[Harald Helfgott]

Wait, I saw the blueprint for the corrected version a few minutes ago, and now I can see only the old blueprint. Where is the new blueprint? (I'm a newbie...)

[Ruben Van de Velde]

The blueprint on the website will be updated after the linked PR is merged

[Harald Helfgott]

Does that mean that I hallucinated the updated blueprint?

[Yaël Dillies]

I think that rather means you read it here: https://github.com/AlexKontorovich/PrimeNumberTheoremAnd/pull/250

[Harald Helfgott]

But how?

[Terence Tao]

Git allows for multiple versions of the repository to exist simultaneously, with the idea being that individual contributors can modify their own copies of the repository as they wish, in order to be merged later into the main repository (subject to approval, review, and continuous integration checks). One can even go back in time and look at old versions of a given repository. It's complicated, but makes a lot of sense once one understands it.