Student Analytic Number Theory

We give power saving asymptotics for \sum_{x,y<X} d(x^4+y^4). This constitutes the first case of estimates for a divisor sums along such a sparse sequence breaching the hyperbola method since 1963 work of Hooley for single variable quadratics. The techniques involve a mix of algebraic number…

Suppose f_1, ..., f_k are functions that have a non-negligible correlation over a product distribution \mu^n; what structure can be deduced about the functions f_1, ..., f_k? While the initial motivation for this problem stemmed from theoretical computer science, and more precisely from…

Given a compact hyperbolic surface of fixed topology, we consider its Laplace eigenvalues together with the structure constants for multiplication with respect to a suitable orthonormal basis of automorphic forms. These numbers obey algebraic constraints analogous to the conformal bootstrap…

A base-g Niven number is an integer divisible by its base-g digit sum. In this talk, we will show that, for any g\geq 3, all sufficiently large integers are the sum of three base-g Niven numbers. The proof uses the circle method, which we also use to count the number of ways to write an integer…

Hindman's conjecture states that for any finite coloring of the integers, there exist natural numbers x and y such that x, y, x+y, xy all have the same color. This conjecture remains open, with its difficulty stemming from the challenge of controlling arithmetic structures that simultaneously…

The form x²+y² covers ½ of the primes, while the forms x²+y², x²+2y² cover ¾ of them. In this talk, we will show that the proportion of primes covered by the forms x²+dy², 1 ≤ d ≤ Δ, is1 - exp((α(Δ) + o(1)) √Δ / log Δ)for some 7π/12 ≤ α(Δ) ≤ 7π/12 + log 4. Furthermore, inspired by…

Let Q be a non-degenerate indefinite quadratic form with at least 3 variables. In the 1980s Margulis proved the longstanding Oppenheim Conjecture, stating that unless Q is proportional to an integral form, the set of values Q attaining at the integer points is dense in R. In Margulis' seminal…

Let n > 1. I will discuss how many of the generalised Fermat equations ax^n + by^n + cz^n = 0 are soluble in the real numbers and all p-adic fields as a,b,c varies. The main new analytic tool is a large sieve inequality for power residue symbols. This talk is based on joint work with Peter…

Szemeredi proved that any subset of the integers with positive upper density contains arbitrarily long arithmetic progressions. Subsequently, Szemeredi's theorem was generalized to the polynomial and multidimensional settings. We will discuss finding the progressions involving rational functions…

For what size random subset D of F_p^n does it hold, with high probability, that any dense subset contains a nontrivial k-term arithmetic progression with common difference in D? We provide a new lower bound on the size of D by showing that a sufficiently small D will be disjoint from a dense…