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Comment by mlechha | Hacker News Reader

I just went through the preprint and I do not understand your comment. What specifically ticked you off? The preprint is well written, arguments are clear and there's enough background for an expert to work things out.

As Atiyah says in the preprint. The magic is the Todd function and the Mathematical framework that comes with it. It seems Atiyah has developed a new framework (which he calls Arithmetic Physics) and a side product of the framework you get a simple proof of RH. I don't know if the proof is correct. But I don't see any signs of crackpottery in the preprint.

Finally, this is in the style of Atiyah. He is known to be a "theory builder" rather than a "problem solver". True to that, he's claiming a whole new way of looking at number theory. So even if the proof turns out to be false. Mathematicians still get some new ideas.

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No, it is not "well written". I'm no expert in analytic number theory, but here are some sanity checks:

His definition of the critical strip (2.4) is wrong.

He works with some family of polynomial functions who agree on the sets K[a] that have open interior (2.1). Of course, two polynomials that agree on infinitely many points are identical. So there really is not much to his "Todd-function". It is just a polynomial.

From his claims 2.3 and 2.4 then follows T(n)=n, for all natural n and hence T(s)=s, as T is a polynomial.

What does "T is compatible with any analytic formula" in (2.4) even mean? Does it mean "for f(X) a everywhere converging power series, then T(f(s))=f(T(s)), for s in C"? This can only hold for T(s)=s, again. So maybe it means something else? He applies it to f(X)=Im(X-1/2), which is not a power series, so what does he mean?

The Hirzebruch reference is a 250pp book. The paragraph on Todd-Polynomials (which are a family of multivariate polynomials, btw. There is no "Todd-polynomial" T in Hirzebruch!) does not contain a formula as claimed in (2.6).

Considering the last two breakthrough claims, that Atiyah made (no complex S^6 sphere and a new proof of Feit-Thompson) vanished in thin air, I remain more than sceptical that this "preprint" can be salvaged.

> He works with some family of polynomial functions who agree on the sets K[a] that have open interior (2.1). Of course, two polynomials that agree on infinitely many points are identical.

Consider f(x, y) := xy and g(x, y) := xy^2

Fixing x=0 note that f and g agree along {(x, y) | x = 0, y in R}. But f is not identical to g. There is no open subset of R^2 such that f and g agree throughout the subset.

Would rewording "two polynomials that agree on infinitely many points are identical" as "two polynomials that agree on any open set" fix this? Or restrict the statement to polynomials in one variable only?

C is not R^2. Neither f nor g is a polynomial over C, which is why they aren't a counterexample.

And x = 0 is not an open set.

agreed, x=0 is not an open set for n>1 dimensions, but it is "infinitely many points"

You're right, I was thinking about univariate polynomials. The "preprint" is only concerned with functions of one complex argument, so this should suffice.

You'll have to go to the previous paper[1], where this Todd function is supposedly defined and...it's not defined there either. I mean, the whole thing is really sad and not something that I feel like talking much. Just give things a read.

[1] https://drive.google.com/open?id=1WPsVhtBQmdgQl25_evlGQ1mmTQ...

Atiyah says, Hirzebruch named and defined the function. Why not look for it in the work of Hirzebruch that Atiyah cites at the end?

Did you do an undergraduate math degree? There are some very real issues with the Todd function, such as no polynomial behaving the way he suggests for pretty elementery reasons. This doesn't even come close to meeting the standards of what a paper should look like, especially one that is claiming such a big result.

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