| 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. |
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?