Our graduate students
Bujold, Crystel
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Faculty of Arts and Science - Department of Mathematics and Statistics
André-Aisenstadt
Courriels
Courses
- MAT1903 H - Calcul différentiel
Research area
Student supervision Expand all Collapse all
Long large character sums
Theses and supervised dissertations / 2019-12
Bujold, Crystel
Abstract
Abstract
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.
Formes quadratiques ternaires représantant tous les entiers impairs
Theses and supervised dissertations / 2013-11
Bujold, Crystel
Abstract
Abstract
In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine
whether a given quadratic form represents all positive integers ; the 15-theorem. In this
thesis, we investigate an analogous problem, that is the search for a similar criterion
allowing one to detect if a quadratic form in three variables represents all odd integers.
We start with a general introduction to the theory of quadratic forms, namely in two
variables, then, we expose different points of view under which quadratic forms can be
considered. We then describe the 15-theorem and its generalizations, with a particular
emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a
proof of two theorems which provide a criteria to determine whether a ternary quadratic
form represents all odd integers.