Scientific

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2}(e^2 −5)\log T$ as $T\rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer

The Farey sequence $\mathcal{F}_Q$ of order $Q$ is the ascending sequence of fractions $\frac{a}{b}$ in the unit interval $(0, 1]$ with $gcd(a, b) = 1$ and $0 < a \leq b \leq Q$. The study of the Farey fractions is of major interest because of their role in problems related to Diophantine approximation. Also, there is a connection between the distribution of Farey fractions and the Riemann hypothesis, which further motivates their study. In this talk, we will discuss the distribution of Farey fractions with some divisibility constraints on denominators by studying their pair-correlation measure. This is based on joint work with Sneha Chaubey.

Recently Martin, Mossinghoff, and Trudgian investigated comparative number theoretic results for a family of arithmetic functions called “fake $\mu$’s”. In their paper, they focused on the bias and oscillation of the summatory function of a fake $\mu$ at the $\sqrt{x}$ scale, while acknowledging that a function with no bias at this scale could well see one at a smaller scale. In this spirit, I will discuss some new oscillation results for the summatory functions of general fake $\mu$’s. This is joint work with Greg Martin.

Please note, this recording is incomplete due to a problem with the room system.

Central to comparative number theory is the study of the difference $\Delta(t) = \pi(t) − li(t)$, where $\pi(t)$ is the prime counting function and $li(t)$ is the logarithmic integral. Prior to a celebrated 1914 paper of Littlewood, it was believed that $\Delta < 0$ for all $t > 2$. We now know however that $\Delta(t)$ changes sign infinitely often, with the first sign change occuring before 10320. Despite this, it still appears that $\Delta(t)$ is negative “on average”, in that integrating $\Delta (t)$ from $t = 2$ onwards yields a negative value. In this talk, we will explore this idea in detail, discussing links with the Riemann hypothesis and also extending such ideas to other differences involving arithmetic functions.

Based on work previously done by Gonek, Graham, and Lee, we show that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with k-fold convolution of von Mangoldt function and the generalized von Mangoldt function. For each $k \in\mathbb{N}$, we study two types of twisted sums:

1. $\sum_{n\leq x} \Lambda^k(n)n^{-iy}$, where $\Lambda^k(n) = \underbrace{\Lambda\star\cdots\Lambda}_\text{k copies}$
2. $\sum_{n\leq x} \Lambda_k(n)n^{-iy}$, where $\Lambda_k(n) :=\sum_{d|n}\mu(d)\left(\log{\frac{n}{d}}\right)^k$.

Where $\Lambda$ is the von Mangoldt function and $\mu$ is the Möbius function, and establish similar connections with RH.