Number Theory

This talk was given as a guest lecture for the PIMS Network Wide Course Analytic Number Theory II in the 2022-2023 academic year.

A directed variant of the famous Oberwolfach problem, the directed Oberwolfach problem considers the following scenario. Given $n$ people seated at $t$ round tables of size $m_1,m_2,\ldots,m_t$, respectively, such that $m_1+m_2+\cdots +m_t=n$, does there exist a set of $n−1$ seating arrangements such that each person is seated to the right of every other person precisely once? I will first demonstrate how this problem can be formulated as a type of graph-theoretic problem known as a cycle decomposition problem. Then, I will discuss a particular style of construction that was first introduced by R. HÄggkvist in 1985 to solve several cases of the original Oberwolfach problem. Lastly, I will show how this approach can be adapted to the directed Oberwolfach problem, thereby allowing us to obtain solutions for previously open cases. Results discussed in this talk arose from collaborations with Andrea Burgess, Peter Danziger, and Daniel Horsley.

For a number field $K$, the P\'olya group of $K$, denoted by $Po(K)$, is the subgroup of the ideal class group of $K$ generated by the classes of the products of maximal ideals of $K$ with the same norm. In this talk, after reviewing some results concerning $Po(K)$, I will generalize this notion to the relative P\'olya group $Po(K/F)$, for $K/F$ a finite extension of number fields. Accordingly, I will generalize some results in the literature about P\'olya groups to the relative case. Then, due to some essential observations, I will explain why we need to modify the notion of the relative P\'olya group to the Ostrowski quotient $Ost(K/F)$ to get a more 'accurate' generalization of $Po(K)$. The talk is based on a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute For Research In Fundamental Sciences).

In 1973, assuming the Riemann hypothesis (RH), Montgomery studied the vertical distribution of zeta zeros, and conjectured that they behave like the eigenvalues of some random matrices. We will discuss some models for zeta zeros starting from the random matrix model but going beyond it and related questions, conjectures and results on statistical information on the zeros. In particular, assuming RH and a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we will discuss our proof of a conjecture of Berry (1988) for the number variance of zeta zeros, in a regime where random matrix models alone do not accurately predict the actual behavior (based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).

Subconvexity problems have maintained extreme interest in analytic number theory for decades. Critical barriers such as the convexity, Burgess, and Weyl bounds hold particular interest because one usually needs to drastically adjust the analytic techniques involved in order to break through them. It has recently come to light that shifted Dirichlet series can be used to obtain subconvexity results. While these Dirichlet series do not admit Euler products, they are amenable to study via spectral methods. In this talk, we construct a shifted multiple Dirichlet series (MDS) and leverage its analytic continuation via spectral decompositions to obtain the Weyl bound in the conductor-aspect for the L-function of a holomorphic cusp form twisted by an arbitrary Dirichlet character. This improves upon the corresponding bound for quadratic characters obtained by Iwaniec-Conrey in 2000. This work is joint with Jeff Hoffstein, Nikos Diamantis, and Min Lee.