# Number Theory

## On the vertical distribution of the zeros of the Riemann zeta-function

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).

## Discrete mathematics in continuous quantum walks

Let $G$ be a graph with adjacency matrix $A$. A continuous quantum walk on $G$ is determined by the complex unitary matrix $U(t)=\exp(itA)$, where $i^2=−1 and $t$ is a real number. Here, $G$ represents a quantum spin network, and its vertices and edges represent the particles and their interactions in the network. The propagation of quantum states in the quantum system determined by $G$ is then governed by the matrix $U(t)$. In particular, $|U(t)_{u,v}|^2$ may be interpreted as the probability that the quantum state assigned at vertex $u$ is transmitted to vertex $v$ at time $t$. Quantum walks are of great interest in quantum computing because not only do they produce algorithms that outperform classical counterparts, but they are also promising tools in the construction of operational quantum computers. In this talk, we give an overview of continuous quantum walks, and discuss old and new results in this area with emphasis on the concepts and techniques that fall under the umbrella of discrete mathematics.

## Remarks on a formula of Ramanujan

In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek (University of Rochester).

## Explicit zero-free regions or the Riemann zeta-function for large t

A zero-free region of the Riemann zeta-function is a subset of the

complex plane where the zeta-function is known to not vanish. In this talk we

will discuss various computational and analytic techniques used to enlarge the

zero-free region for the Riemann zeta-function, when the imaginary part of a

complex zero is large. We will also explore the limitations of currently known

approaches. This talk will reference a number of works from the literature,

including a joint work with M. Mossinghoff and T. Trudgian.

## 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.