Student Analytic Number Theory

The polynomial Szemer\'edi theorem of Bergelson and Leibman gives broad conditions under which polynomial patterns must appear in every positive-density subset of Z^d. When the polynomials do not vanish at zero, the correct replacement condition is conjecturally joint intersectivity, a…

We present solutions to two problems on indefinite integral ternary quadratic forms. The first, highlighted by Margulis in 1990, concerns the distribution of the ternary Markoff spectrum associated with minima of forms. The second, initiated by Serre in 1990, is on the asymptotic growth of the…

How many rational points with a denominator of a given size lie inside a non-isotropic neighborhood of a manifold of arbitrary dimension? This talk presents recent progress toward answering this question, based on joint work with Sam Chow, Niclas Technau, and Han Yu. Our results resolve the…

We will talk about new phenomena observed when studying the first moment of cubic L-functions at any s in the critical strip. We show that there is a phase transition in the moment at s=1/3, and an interesting symmetry in the moment happens between s>1/3 and s<1/3. In particular, at s=1/2…

In this talk, I will focus on simultaneous non-vanishing results for Dirichlet L-functions at the central point 1/2. Specifically, I will describe how to obtain a positive proportion of simultaneous non-vanishing result for four L-functions as we vary over characters \chi modulo q,…

Let A be a finite subset of an abelian group which is "approximately closed under addition" in the sense that |A+A| < K|A|. To what extent is A approximately algebraically structured? We show that A can be covered efficiently by translates of subgroups and sets arising from convex bodies,…

In this talk, I’ll discuss a question concerning sign changes in partial sums of random multiplicative functions. This is joint work with R. Angelo, M. Aymone, O. Klurman, and M. Xu.

We show that the least natural number having an odd number of prime factors and belonging to any arithmetic progression $a \pmod q$ is bounded by $q^{2+o(1)}$. This can be seen as a multiplicative analogue of Linnik's problem on the least prime in an arithmetic progression. This is joint work…

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…