MAT6627: the distribution of prime numbers
Université de Montréal, Fall 2016
Contact information
Instructor: Dimitris Koukoulopoulos
Email: koukoulo AT dms.umontreal.ca
Office: 4163 Pav. André Aisenstadt

Schedule
Tuesday 10h30  11h30 and Friday 10h30  12h30 (4186 Pav. Andre Aisenstadt)
The course starts on Friday September 2. There is no course during the weeks of Monday October 24 and of Monday November 7. There will be extra hours of teaching on some Tuesdays from 11h30 to 12h30 to cover the loss during the week of November 7.
Textbook and references
I will follow my own notes that can be found here.
Some other useful references are:
 H. L. Montgomery and R. C. Vaughan,
Multiplicative number theory. I. Classical theory.
Cambridge Studies in Advanced Mathematics, 97. Cambridge University Press, Cambridge, 2007.
 H. Iwaniec et E. Kowalski,
Analytic number theory.
American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004.
 G. Tenenbaum,
Introduction to analytic and probabilistic number theory.
Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995.
 H. Davenport,
Multiplicative number theory. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. SpringerVerlag, New York, 2000.
 T. Apostol, Introduction to analytic number theory. Undergraduate Texts in Mathematics. SpringerVerlag, New YorkHeidelberg, 1976.

My notes on sieve methods.
You can also consult thenotes of A. J. Hildebrand.
Content
The course will cover the following subjects:
 Arithmetic functions and techniques of estimation of their partial sums
 Elementary prime number theory
 Introduction to sieve methods
 Dirichlet series and complexanalytic methods
 The prime number theorem
 Dirichlet characters
 The prime number theorem for arithmetic progressions
 The large sieve and the BombieriVinogradov theorem
 Bounded gaps between prime numbers
If there is sufficient time, we will also cover Linnik's theorem and the analytic theory of Lfunctions.
Evaluation
Your grade will be determined by your performance in three homework sets and one final project, explained below. Here is the breakdown:
Evaluation

Percentage

Homework 1

20%

Homework 2

25%

Homework 3

25%

Final project: oral part

10%

Final project: written part

20%

Homework
You may discuss the homework exercises with your peers, but you must write your solutions to them on your own. For the solution of the homework problems, you may consult only my course notes as well as the ones you have taken during the class. Lastly, you must write your solutions to the homework sets in LaTeX and send them to me by email before the deadline.
Homework 1 (due 23 septembre) :
PDF TeX
Homework 2 (due 21 octobre) :
PDF TeX
Homework 3 (due 2 decembre) :
PDF TeX
Final project
Your final project will be based on an topic of your choice from the list below. You must give a written presentation of this topic (510 pages in LaTeX, including either complete proofs of sketches, depending on the lenght) and do an oral presentation of 30 minutes. orale de 30 minutes. The written part is due December 9 (do not leave this for the last moment!) and the oral presentations will take place during the week of December 12.
 Selberg, Atle;
Note on a paper by L. G. Sathe.
J. Indian Math. Soc. (N.S.) 18, (1954). 83–87.

Linnik's theorem: chapter 7 of the course notes.

The analytic theory of Lfunctions: chapter 10 of the course notes.

Granville, Andrew; Soundararajan, K. Large character sums: pretentious characters and the PólyaVinogradov theorem. J. Amer. Math. Soc. 20 (2007), no. 2, 357–384.

Hildebrand, Adolf(1IL); Tenenbaum, Gérald(FNANC)
On integers free of large prime factors.
Trans. Amer. Math. Soc. 296 (1986), no. 1, 265–290.

Elementary proof of the prime number theorem:
 Selberg, Atle An elementary proof of the primenumber theorem. Ann. of Math. (2) 50, (1949). 305–313. (Reviewer: A. E. Ingham) 10.0X
 Erdos, P.
On a new method in elementary number theory which leads to an elementary proof of the prime number theorem.
Proc. Nat. Acad. Sci. U. S. A. 35, (1949). 374–384.
 Ternary Goldbach conjecture: chapter 26 of the book by Davenport cited above.
 The halfdimensional sieve :
 Iwaniec, H.
The half dimensional sieve.
Acta Arith. 29 (1976), no. 1, 69–95.
 Chapitre 14 du livre "Opera de cribro"; Friedlander, John; Iwaniec, Henryk, American Mathematical Society Colloquium Publications, 57. American Mathematical Society, Providence, RI, 2010.
 Means of general multiplicative functions: the theorems of Wirsing, Delange and Halász. See chapter III.4 of the book by Tenenbaum cited above.
 Matomaki, Kaisa; Radziwill, Maksym; Multiplicative functions in short intervals. Ann. of Math. (2) 183 (2016), no. 3, 1015–1056.

Zerodensity estimates of Lfunctions. See chapter 10 of the book by IwaniecKowalski cited above.
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