Number Theory

This talk will begin with a study on explicit bounds for $\psi(x)$ starting with the work of Rosser in 1941. It will also cover various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, Platt-Trudgian, Büthe, and Fiori-Kadiri-Swidinsky. In the second part of this talk, I will provide an overview of my master's thesis which is a survey on 'Estimating $\pi(x)$ and Related Functions under Partial RH Assumptions' by Jan Büthe. This article provides the best known bounds for $\psi(x)$ for small values of $x$ in the interval $[e^{50},e^{3000}]$. A distinctive feature of this paper is the use of Logan's function and its Fourier Transform. I will be presenting the main theorem in Büthe's paper regarding estimates for $\psi(x)$ with other necessary results required to understand the proof.

In the 1960s Shanks conjectured that the $\zeta\'(\rho)$, where $\rho$ is a non-trivial zero of zeta, is both real and positive in the mean. Conjecturing and proving this result has a rich history, but efforts to generalise it to higher moments have so far failed. Building on the work of Keating and Snaith using characteristic polynomials from Random Matrix Theory, the Hybrid model of Gonek, Hughes and Keating, and the Ratios Conjecture of Conrey, Farmer, and Zirnbauer, we have been able to produce new conjectures for the full asymptotics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.

Graphical and Digraphical Regular Representations (GRRs and DRRs) are a concrete way to visualise the regular action of a group, using graphs. More precisely, a GRR or DRR on the group G is a (di)graph whose automorphism group is isomorphic to the regular action of G on itself by right-multiplication.

For a (di)graph to be a DRR or GRR on G, it must be a Cayley (di)graph on G. Whenever the group G admits an automorphism that fixes the connection set of the Cayley (di)graph setwise, this induces a nontrivial graph automorphism that fixes the identity vertex, which means that the (di)graph is not a DRR or GRR. Checking whether or not there is any group automorphism that fixes a particular connection set can be done very quickly and easily compared with checking whether or not any nontrivial graph automorphism fixes some vertex, so it would be nice to know if there are circumstances under which the simpler test is enough to guarantee whether or not the Cayley graph is a GRR or DRR. I will present a number of results on this question.

This is based on joint work with Dave Morris and with Gabriel Verret.

Peter Borewein empirically discovered quite a number of mysteries involving sign patterns of coefficients of polynomials of the form $f_{p,s,n}(q):=\prod_{j=0}^{n} \prod_{k=1}^{p-1} (1-q^{pj+k})^{s}$ ($p$ a prime and $s,n \in \mathbb{N}$). In the case $(p,s) \in \{(3,1), (3,2)\}$, he conjectured that the coefficients follow a repeating + - - pattern, and in the case $(p,s)=(5,1)$, it was conjectured that the coefficients follow a repeating + - - - - sign pattern. We consider a weaker problem of finding the signs of partial sums of coefficients along some arithmetic progressions. We use a combinatorial sieving principle by Li-Wan and elementary character theory to asymptotically estimate and find the signs of these partial sums. We find that the signs of these partial sums are compatible with the sign pattern in Borewein's conjectures. This is based on joint work with Ankush Goswami.

We establish a Weyl-type estimate for exponential sums over irreducible elements in function fields. As an application, we generalize an equidistribution theorem of Rhin. Our estimate works for polynomials with degree higher than the characteristic of the field, a barrier to the traditional Weyl differencing method. In this talk, we briefly introduce Lê-Liu-Wooley's original argument for ordinary Weyl sums (taken over all elements), and how we generalize it to estimate bilinear exponential sums with general coefficients. This is joint work with Jérémy Campagne (Waterloo), Thái Hoàng Lê (Mississippi) and Yu-Ru Liu (Waterloo).