Student Analytic Number Theory

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…

The divisor function $d(n)$ that counts the number of divisors of a given integer $n$ is one of the central functions in analytic number theory. Despite its simplicity, it is easy to formulate questions about it that are out of reach with current methods, for example when considering its…

Abstract:  A classical technique to upper bound ranks of elliptic curve is descent. The case of 2-descent, when studied for quadratic twist families of elliptic curves has attracted the attention of several authors, Heath-Brown, Friedlander--Iwaniec--Mazur--Rubin, Kane, Smith, among some of…

Abstract

Let a,b>0 be coprime integers. Assuming a conjecture on Hecke eigenvalues along binary cubic forms, we prove an asymptotic formula for the number of primes of the form ax^2+by^3 with x < X^(1/2) and y  < X^(1/3). The proof combines sieve methods with the theory of real quadratic…

We will finish our discussion of recent work of Guth and Maynard on large values of Dirichlet polynomials.