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Koukoulopoulos, Dimitrios

Full Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-6716

Affiliations

• Membre Centre de recherches mathématiques
• Membre CRM  Centre de recherches mathématiques

Student supervision Expand allCollapse all

Primes with a missing digit : distribution in arithmetic progressions and sieve-theoretic applications Theses and supervised dissertations / 2021-07
Nath, Kunjakanan
Abstract
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.

Long large character sums Theses and supervised dissertations / 2019-12
Bujold, Crystel
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.

Linnik's theorem : a comparison of the classical and the pretentious approach Theses and supervised dissertations / 2018-12
Matte, Joelle
Abstract
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.

Anatomy of smooth integers Theses and supervised dissertations / 2017-07
Abstract
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.

La distribution des zéros des fonctions L Theses and supervised dissertations / 2016-08
Comeau-Lapointe, Antoine
Abstract
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.

Selected publications Expand allCollapse all

Sieve weights and their smoothings

A. Granville, D. Koukoulopoulos and J. Maynard, Sieve weights and their smoothings , (2016), , preprint

Permutations contained in transitive subgroups

S. Eberhard, K. Ford and D. Koukoulopoulos, Permutations contained in transitive subgroups , 33 pages (2016), arXiv:1605.01068, preprint

The frequency of elliptic curve groups over prime finite fields

V. Chandee, C. David, D. Koukoulopoulos and E. Smith, The frequency of elliptic curve groups over prime finite fields , 41 pages (2016), DOI: 10.4153/CJM-2015-013-1, arXiv:1405.6923, Canad. J. Math

When the sieve works

A. Granville, D. Koukoulopoulos and K. Matom\"aki, When the sieve works 164, no. 10, 1935-1969 (2015), DOI:10.1215/00127094-3120891, arXiv:1205.0413 , Duke Math. J.

The cardinality of sumsets: different summands

B. Murphy, E. A. Palsson et G. Petridis, The cardinality of sumsets: different summands 167, no. 4, 375-395 (2015), , Acta Arith.

The mean-value of a product of shifted multiplicative functions and the average number of points of elliptic curves

R. Balasubramanian and S. Giri, The mean-value of a product of shifted multiplicative functions and the average number of points of elliptic curves 157, 37-53 (2015), , J. Number Theory

\href{https://dx.doi.org/10.1007/978-3-319-22240-0}{Best possible densities, as a consequence of Zhang-Maynard-Tao}

A. Granville, D. M. Kane, D. Koukoulopoulos et R. Lemke Oliver, \href{https://dx.doi.org/10.1007/978-3-319-22240-0}{Best possible densities, as a consequence of Zhang-Maynard-Tao} Analytic Number Theory, In Honor of Helmut Maier's 60th Birthday, Springer, New York, 133-144 (2015), , C. Pomerance et M. Th. Rassias

Sums of Euler products and statistics of elliptic curves

C. David, D. Koukoulopoulos and E. Smith, Sums of Euler products and statistics of elliptic curves , 56 pages (2015), arXiv:1510.05935, preprint

Sieve methods and applications

D. Koukoulopoulos, Sieve methods and applications , In progress (2015), ,

Best possible densities, as a consequence of Zhang-Maynard-Tao

A. Granville, D. M. Kane, D. Koukoulopoulos and R. Lemke Oliver, Best possible densities, as a consequence of Zhang-Maynard-Tao , 133-144 (2015), arxiv:1410.8198, Analytic Number Theory, In Honor of Helmut Maier's 60th Birthday, Springer, New York

Primes in short arithmetic progressions

D. Koukoulopoulos, Primes in short arithmetic progressions 11, no. 5, 1499-1521 (2015), arXiv:1405.6592, DOI:10.1142/S1793042115400035, Int. J. Number Theory

The frequency and the structure of large character sums

J. Bober, L. Goldmakher, A. Granville and D. Koukoulopoulos, The frequency and the structure of large character sums , 58 pages (2014), arXiv:1410.8189, preprint

On the number of integers in a generalized multiplication table

D. Koukoulopoulos, On the number of integers in a generalized multiplication table 689, 33-99 (2014), arXiv:1102.3236, DOI:10.1515/crelle-2012-0064 , J. Reine Angew. Math.

On the concentration of certain additive functions

D. Koukoulopoulos, On the concentration of certain additive functions 162, no. 3, 223-241 (2014), arXiv:1111.1040, DOI:10.4064/aa162-3-2 , Acta Arith.

Group structures of elliptic curves over finite fields

V. Chandee, C. David, D. Koukoulopoulos and E. Smith, Group structures of elliptic curves over finite fields no. 19, 5230-5248 (2014), DOI: 10.1093/imrn/rnt120, arXiv:1210.3880 , Int. Math. Res. Not. IMRN

On multiplicative functions which are small on average

D. Koukoulopoulos, On multiplicative functions which are small on average 23 no. 5, 1569-1630 (2013), DOI: 10.1007/s00039-013-0235-6, arXiv:1111.2659 , Geom. Funct. Anal.

Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

D. Koukoulopoulos, Pretentious multiplicative functions and the prime number theorem for arithmetic progressions 149, no. 7, 1129-1149 (2013), DOI: 10.1112/S0010437X12000802, arXiv:1203.0596, Compos. Math.

Arrangements of stars on the American flag

D. Koukoulopoulos, J. Thiel, Arrangements of stars on the American flag 119, no. 6, 443-450 (2012), 10.4169/amer.math.monthly.119.06.443, Amer. Math. Monthly

Divisors of shifted primes

D. Koukoulopoulos, Divisors of shifted primes no. 24, 4585-4627 (2010), DOI: 10.1093/imrn/rnq045, arXiv:0905.0163 , Int. Math. Res. Not. IMRN

Localized factorizations of integers

D. Koukoulopoulos, Localized factorizations of integers (3) 101, no. 2, 392-426 (2010), DOI: 10.1112/plms/pdp056, arXiv:0809.1072 , Proc. London Math. Soc.

A reciprocity theorem for certain hypergeometric series

B. C. Berndt, D. Koukoulopoulos, A reciprocity theorem for certain hypergeometric series 137, 2369-2373 (2009), 10.1090/S0002-9939-09-09777-9, Proc. Amer. Math. Soc.