Power-free values, large deviations, and integer points on irrational curves

Let $f\in \mathbb{Z}\lbrack x\rbrack$ be a polynomial of degree $d\geq 3$ without roots of multiplicity $d$ or $(d-1)$. Erd\H{o}s conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.Comment: 39 pages; rather major revision, with strengthened and generalized statement

The ternary Goldbach conjecture is true

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualitative gains. The improved estimates on exponential sums are proven in the author's papers on major and minor arcs for Goldbach's problem. One of the highlights of the present paper is an optimized large sieve for primes. Its ideas get reapplied to the circle method to give an improved estimate for the minor-arc integral.Comment: 79 pages, 1 figure. Minimal change