In 1972, Billingsley proved that the sequence of relative sizes of prime factors of a random integer $N_x$ chosen uniformly from $\mathbb{N} \cap [1, x]$ converges to a Poisson--Dirichlet process $(V_i)_{i \ge 1}$ with parameter 1. In 2002, Arratia constructed a coupling between $N_x$ and $(V_i)_{i \ge 1}$ such that the sizes of the prime factors of $N_x$ are close to $(V_i)_{i \ge 1}$ in $\ell^1$ distance. He further conjectured that this coupling could be improved, and proposed a conjecture on the optimal $\ell^1$ distance achievable by a coupling of these objects. In the first part of this thesis, we prove this conjecture: there exists a coupling of $N_x$ and $(V_i)_{i \ge 1}$ such that for all $x \ge 2$, $\mathbb{E} \sum_{i\ge 1} |\tfrac{\log P_i}{\log x} -V_i| \asymp \frac{1}{\log x},$ where $(P_i)_{i \ge 1}$ is the unique non-increasing sequence of primes or ones satisfying $N_x = \prod_{i\ge 1} P_i$. We also generalize this coupling to the case where $N_x$ follows a multiplicative distribution belonging to a certain class. The second part of the thesis is devoted to three applications of this coupling. First, we derive an asymptotic formula for the average size of the middle divisor of an integer. Second, we give an asymptotic formula for the number of integers $\le x$ that have a divisor in a given interval $(y, z]$ when $\frac{z}{y} \to \infty$ as $x \to \infty$. Finally, we show that when randomly selecting a $k$-factorization of a random integer in a certain way, the sizes of the factors follow a Dirichlet law, thereby improving a result of Leung (2023).

This thesis consists of two projects. The first one is concerned with the structure of multiplicative functions whose averages are small. In particular, in this project, we establish the average behaviour of the prime values \(f(p)\) for suitable multiplicative functions \(f\) when their partial sums \(\sum_{n\leqslant x}f(n)\) admit logarithmic cancellations over their trivial upper bound. This result extends previous related work of Koukoulopoulos and Soundararajan and it is built upon ideas coming from the more careful treatment of Koukoulopoulos on the special case of bounded multiplicative functions. The second project of the dissertation is inspired by an analogue of an estimate that Linnik deduced in his attempt to prove his celebrated theorem regarding the size of the smallest prime number of an arithmetic progression. This estimate provides a strongly uniform asymptotic formula for the sums of the von Mangoldt function \(\Lambda\) on arithmetic progressions. In the literature, its existing proofs involve non-trivial information about the zeroes of Dirichlet \(L\)-functions \(L(\cdot,\chi)\) and the purpose of the second project is to present a different, more elementary approach which recovers this estimate by avoiding the “language” of those zeroes. For the development of this alternative method, we make use of ideas that appear in the pretentious large sieve of Granville, Harper and Soundararajan. Moreover, as in the case of the first project, we also borrow insights from the work of Koukoulopoulos on the structure of bounded multiplicative functions with small averages.

The lonely runner conjecture was formulated by J.M. Wills en (1972) and Thomas Cusick (1973). If ∥·∥ denotes the distance from integers, for x ∈ R ∥x∥ = mink∈Z(|x − k|), this conjecture is asking whether or not for any set of n + 1 distinct real numbers {v1,v2 . . . vn+1} and for any k ∈ {1,2,3 . . . ,n + 1} there is a time t ∈ R such that for any other speed vi,i ̸= k we have ∥t(vi − vk)∥ ≥ 1 n+1 . It has been proven to be true for n + 1 ≤ 7 , the last case n + 1 = 7 was shown by Barajas and Serra in 2007. Many authors have wrote about this subject each bringing more knowledge. In this thesis, there will be an exposure on different techniques that have been used to prove the cases for n + 1 ≤ 7, differents cases in wich the conjecture holds and the problem of getting better lower bounds for the gap of loneliness.

A covering system is a finite set of arithmetic progressions with the property that every integer belongs to at least one of them. The study of covering systems was started by Erdős in the 1950’s, and he asked many questions about them in the following years. One of the most famous questions he asked was if the minimum modulus of a covering system with distinct moduli is bounded uniformly. In 2015, Hough showed that it is at most 1016. Following on his work, but simplifying the method, Balister, Bollobás, Morris, Sahasrabudhe and Tiba showed that it is at most 616, 000. Their method led them to many further applications. Notably, they counted the number of covering systems with a fixed number of moduli. The first part of this thesis seeks to study a related question, that is to count the number of covering systems with a given set of moduli. The technique developped to do this for some sets will lead us to look at symmetries of covering systems. The second part of this thesis will look at variants of the minimum modulus problem. Notably, we will be looking at bounds on the minimum modulus of a covering system of multiplicity s, that is a covering system in which each moduli appears at most s times, as well as bounds on the minimum modulus of a covering system of multiplicity 1 of an arithmetic progression, and finally look at bounds for the n-th smallest modulus in a covering system.

Let q ⩾ 2 be a fixed prime power. The main objective of this thesis is to study the asymptotic behaviour of the arithmetic function Π_q(n,k) counting the number of monic polynomials that are of degree n and have exactly k irreducible factors (with multiplicity) over the finite field F_q. Warlimont and Car showed that the object Π_q(n,k) is approximately Poisson distributed when 1 ⩽ k ⩽ A log n for some constant A > 0. Later Hwang studied the function Π_q(n,k) for the full range 1 ⩽ k ⩽ n. We will first prove an asymptotic formula for Π_q(n,k) using a classical analytic technique developed by Sathe and Selberg. We will then reproduce a simplified version of Hwang’s result using the Sathe-Selberg formula in the function field. We also compare our results with the analogous existing ones in the integer case, where one studies all the natural numbers up to x with exactly k prime factors. In particular, we show that the number of monic polynomials grows at a surprisingly higher rate when k is a little larger than logn than what one would speculate from looking at the integer case. To present the above work, we first start with basic analytic number theory in the context of polynomials. We then introduce the key arithmetic functions that play a major role in our thesis and briefly discuss well-known results concerning their distribution from a probabilistic point of view. Finally, to understand the key results, we give a fairly detailed discussion on the function field analogue of the Sathe-Selberg formula, a tool recently developed by Porrit and subsequently use this tool to prove the claimed results.

The theme of this thesis is to understand the distribution of prime numbers, which is a central topic in analytic number theory. More precisely, we prove Bombieri-Vinogradov type theorems for primes with a missing digit in their b-adic expansion for some large positive integer b. The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi-linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form p=1+m^2+n^2 with a missing digit in a large odd base b.

Under the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, Rubinstein and Sarnak proved that the values of x which have more prime numbers less than or equal to x of the form 4n + 3 than primes of the form 4n + 1 have a logarithmic density of approximately 99.59%. In general, the study of the difference #{p < x : p in A} − #{p < x : p in B} for two subsets of the primes A and B is called the prime number race between A and B. In this thesis, we will analyze the prime number race between the primes p such that 2p + 1 is also prime (these primes are called the Sophie Germain primes) and the primes p such that 2p − 1 is also prime. To understand this, we first present Rubinstein and Sarnak’s analysis to understand where the bias between primes that are 1 (mod 4) and the ones that are 3 (mod 4) comes from and give a conjecture on the distribution of Sophie Germain primes.

This thesis deals with a central topic in analytic number theory, namely that of characters and more specifically, that of character sums. More precisely, we will develop a result concerning the maximal value that can be attained by some long character sum. In Chapter 1 are discussed the notions and techniques that will be necessary in the elaboration of the proof of the main result. We will discuss notions of harmonic analysis, classical number theoretic techniques, as well as give an overview of smooth numbers. Chapter 2 will serve as an introduction to the theory pertaining to Dirichlet characters and character sums. Basic properties and classical theorems will be covered and we will provide a survey of recent results closely related to the main topic on interest in this thesis. We will give in Chapter 3 a first result which will lead this thesis to diverge into the field of lattices. It comes up as an auxiliary result to the main result, but bares an interest independent to characters. We will discuss the order of magnitude of multiples of a chosen lattice vector, when the multipliers lie in prescribed congruence classes. Chapter 4 will serve as a bridge between lattices and characters and we will study the consequences of applying the theorems we proved in Chapter 3 to characters. We will derive results that will be key to the proof of our main theorem. In Chapter 5, we will prepare the ground for the proof of our main theorem by unveiling some preliminary estimates that will be needed. In particular, the chapter will consist of two parts: one treating of exponential sums, while the other one will be concerned with smooth numbers. Finally, Chapter 6 will be the apex of this thesis and will provide the proof of our main result on character sums. The argument built in this chapter will allow us to prove a lower bound for the maximal value that can be reached by a character among the characters modulo a prime number q.

The goal of this master's thesis is to understand Linnik's theorem, which gives us an upper bound for the first prime number in an arithmetic progression. We will analyze and compare two distinct methods: the classical approach and the pretentious approach. The first one relies on zeros of Dirichlet L-functions. The second one is based on Halász's theorem and distance functions. It was developped by Granville annd Soundarajan.

The object of the first chapter of this thesis is to review the materials and tools in analytic number theory which are used in following chapters. We also give a survey on the development concerning the number of y−smooth integers, which are integers free of prime factors greater than y. In the second chapter, we shall give a brief history about a class of arithmetical functions on a probability space and we discuss on some well-known problems in probabilistic number theory. We present two results in analytic and probabilistic number theory. The Erdos multiplication table problem asks what is the number of distinct integers appearing in the N × N multiplication table. The order of magnitude of this quantity was determined by Kevin Ford (2008). In chapter 3 of this thesis, we study the number of y−smooth entries of the N × N multiplication. More concretely, we focus on the change of behaviour of the function A(x,y) in different ranges of y, where A(x,y) is a function that counts the number of distinct y−smooth integers less than x which can be represented as the product of two y−smooth integers less than p x. In Chapter 4, we prove an Erdos-Kac type of theorem for the set of y−smooth integers. If !(n) is the number of distinct prime factors of n, we prove that the distribution of !(n) is Gaussian for a certain range of y using method of moments.

The Katz and Sarnak philosophy states that the distribution laws of zeros of $L$-functions follow the distribution laws of eigenvalues of random matrices. The zeros near the central point would reveal the symmetry type of our family of $L$-functions. Once the symmetry has been identified, it is conjectured that many statistics associated to the zeros would be predicted by the eigenvalues of the corresponding group of random matrices. This thesis will study the low-lying zeros of the family of elliptic curves over $\mathbb{Q}[i]$. Brumer computed the symmetry type of the family of elliptic curves over $\mathbb{Q}$ in 1992. New challenges arising from this generalisation over number fields of his work will be revealed in this thesis.