Number Theory

I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo square-free Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment. No such asymptotic formula was previously known for a cubic family (even over function fields).

Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the family of quadratic characters and the family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

Let ord𝑝(𝑎)be the order of 𝑎in (ℤ/𝑝ℤ)∗. In 1927, Artin conjectured that the set of primes 𝑝for which an integer 𝑎≠−1,◻is a primitive root (i.e. ord𝑝(𝑎)=𝑝−1) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk, we will study the behaviour of ord𝑝(𝑎)as 𝑝varies over primes. In particular, we will show, under GRH, that the set of primes 𝑝for which ord𝑝(𝑎)is “𝑘prime factors away” from 𝑝−1− 1 has a positive asymptotic density among all primes, except for particular values of 𝑎and 𝑘. We will interpret being “𝑘prime factors away” in three different ways:
𝑘=𝜔(𝑝−1ord𝑝(𝑎)),𝑘=Ω(𝑝−1ord𝑝(𝑎)),𝑘=𝜔(𝑝−1)−𝜔(ord𝑝(𝑎)).

We will present conditional results analogous to Hooley’s in all three cases and for all integer 𝑘. From this, we will derive conditionally the expectation for these quantities.

Furthermore, we will provide partial unconditional answers to some of these questions.

This is joint work with Leo Goldmakher and Greg Martin.

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).