# Number Theory

## Almost periodicity and large oscillations of prime counting functions

If we assume the relevant Riemann hypotheses, after a suitable rescaling many functions counting certain primes become almost periodic. There are different notion of almost periodicity in use; here we consider the notion induced by the norm $||f|| = \sup_{x∈\mathbb{R}} \int_x^{x+1} |f(t)|^2\,dt$. We show that if a function $f$ can be approximated by linear combinations of periodic functions with respect to this norm, then the level sets $\left\{x: f(x) \geq t\right\}$ are almost periodic for all real $t$ with at most countably many exceptions. We also compare this notion to other notions of almost periodicity in use.

*Please note, the wrong video feed was captured for this lecture so the writing on the blackboard is not legible.*

## A race problem arising from elliptic curves

Given an elliptic curve $E/\mathbb{Q}$, we can consider its trace of Frobenius, denoted as $a_p(E)$, where $p$ is a prime. We will discuss the race problem arising from these ap values and the general strategy in attacking these problems.

## Ramanujan sums and the Hardy–Littlewood prime tuple conjecture

In 1999, Gadiyar and Padma discovered a simple heuristic to derive the generalized twin prime conjecture using an orthogonality principle for Ramanujan sums originally discovered by Carmichael. We derive a limit formula for higher convolutions of Ramanujan sums, generalizing an old result of Carmichael. We then apply this in conjunction with the general theory of arithmetical functions of several variables to give a heuristic derivation of the Hardy–Littlewood formula for the number of prime $k$-tuples less than $x$.

## Number theory versus random matrix theory: the joint moments story

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

## Zero-free regions of the Riemann zeta-function

A zero-free region is a subset of the complex plane where the Riemann zeta-function does not vanish. Such regions have historically been used to further our understanding of prime-number distributions. In the classical approach, we first assume that a zero exists off the critical line, then arrive at an inequality involving its real and imaginary parts. One notable aspect of the classical argument is that it does not require any knowledge about the relationship between the zeroes. However, it is well known that the location of a hypothetical zero depends strongly on the behaviour of nearby zeroes—for instance, N. Levinson showed in 1969 that if zeroes of the zeta-function are well-spaced near the 1-line, then we can obtain a zero-free region stronger than any that are currently known. In this talk we will discuss some ideas on how one might incorporate information about distributions of hypothetical zeroes to improve existing zero-free regions.

## Spacing statistics of the Farey sequence

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.

## Unconditional comparative prime number theory over function fields

In classical comparative prime number theory, it is customary to assume some kind of linear independence hypothesis about the zeros of the underlying L-functions. These hypotheses are completely out of reach of current methods. However, in the function field case, it is sometimes possible to prove them, or at least to show they hold generically. In this talk I will present recent results in comparative prime number theory over function fields that establish infinite families of “irreducible polynomial races” which we can study unconditionally. Some of those results are joint work with L. Devin, D. Keliher, and W. Li.

## Oscillation results for the summatory functions of fake \mu

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.*

## On the generalised Dirichlet divisor problem

In this talk we present new unconditional estimates on $\Delta k(x)$, the remainder term associated with the generalised divisor function, for large $k$. By combining new estimates of exponential sums and Carlson’s exponent, we show that $\Delta k(x) \ll x^{1−1.224(k−2.36)^{−2/3}}$ for $k \geq 58$ and $\Delta k(x) \ll x^{1−1.889k^{−2/3}}$ for all sufficiently large fixed $k$. This is joint work with Andrew Yang.

## Average value of $\pi(t) - li(t)$

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.