The prestige has tempted many mathematicians over the years, none of which has yet been awarded the prize. The details are sketchy to say the least.
No, it is not "well written". Atiyah continued to publish subsequently, including several surveys, a popular book,  and another paper with Segal on twisted K-theory. Mathematicians still get some new ideas.
Or restrict the statement to polynomials in one variable only? While his latest proof has yet to undergo the rigorous peer review process necessary to test its validity, the initial reaction has been one of cautious scepticism.
And somewhere the proof of a conjecture about the zeta function has to mention the zeta function! Atiyah and Bott  showed that this could be deduced from a more general formula in equivariant cohomologywhich was a consequence of well-known localization theorems.
In a series of tweetsUniversity of California, Riverside mathematical physicist John Carlos Baez wrote that he has "huge respect for Atiyah, whose earlier work revolutionized geometry and physics," but predicted that his written proof "will not convince experts.
This allowed him to find an explicit formula for the conformally invariant Green's function on a 4-manifold. The first announcement of the Atiyah—Singer theorem was their paper. It is more convenient to classify instantons on a sphere as this is compact, and this is essentially equivalent to classifying instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant.
By combining their insights, and assuming the Riemann hypothesis does not design thesis architecture, Atiyah claims to reach a sir michael atiyah riemann hypothesis paper contradiction, implying that the hypothesis must in fact be correct.
Atiyah showed  that the moment map was closely related to geometric invariant theoryand this idea was later developed much further by his student F. That I believe is what I have done and, had [Sir Edmund] Hillary and Tenzing Norgay waited, they might have been beaten to their aim by a local shepherd without special mountaineering skills.
Atiyah and the Riemann Hypothesis: What are the next steps? › Heidelberg Laureate Forum With Elmer ReesAtiyah studied the problem of the relation between topological and holomorphic vector bundles on projective space. With Hitchin and Singer  he calculated the dimension of the moduli space of irreducible self-dual connections instantons for any principal bundle over a compact 4-dimensional Riemannian manifold the Atiyah—Hitchin—Singer theorem.
One paper  is a detailed study of the Dedekind eta function from the point of view of topology and the index theorem. For trivial groups G this gives the index theorem, and for a finite group G acting with isolated fixed points it gives the Atiyah—Bott fixed point theorem.
If the manifold is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. And that's just the short version of his achievements.
But its modern reformulation, by German mathematician Bernhard Riemann inhas to do with the location of the zeros of what is now known as the Riemann zeta function.