Analytic Number Theory (topics) 2023/'24
Lecturer prof. G. Molteni
The course is addressed to students in Mathematics at least in their third year degree with a solid knowledge of the basic tools in analysis; some familiarity with the complex analysis will be also useful. The course touches four main topics of the Analytic Number Theory, each one characterized by its proper tools:
| Programme | |
| Topic 1: | The prime number theorem. Riemann Zeta function, zero free region and proof of the result with Bombieri-Wirsing's ``elementary'' argument (giving also a good estimate for the remainder term). |
| Topic 2: | Some results about sieve methods. The Selberg Lambda-square method, Brun-Titchmarsh's theorem and Brun's result about the twin primes density. |
| Topic 3: | Sumsets. Schnirelmann's result about the representability of integers as sum of primes. Results relating Schnirelmann's and the natural densities. Set of integers containing linear sequences: Van der Waerden's Theorem, Szemerédi's proof of the Erdős-Turàn conjecture (sketch). Linear sequences and primes: the result of Green and Tao (very, very roughly sketched). |
| Topic 4: | Waring's problem. The qualitative approach of Linnik and Newmann. |
| Note: | Every suggestion from the students will be welcome, in particular it will be possible to (partially) modify the second part of the program according to their needs and interest. |
| Bibliography | |
| G. Molteni: Notes for the course. | |
| H. Iwaniec, E. Kowalski: Analytic number theory, American Mathematical Society Colloquium Publications 53, American Mathematical Society, Providence RI, 2004. | |
| H. L. Montgomery, R. C. Vaughan: Multiplicative number theory I. Classical theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007. | |
| P. Pollack: Not always buried deep, A second course in elementary number theory, American Mathematical Society, Providence RI, 2009. |
Lectures
| Tuesday | 8.30-10.30 | Room 5 |
| Friday | 13.30-15.30 | Room 5 |
Exam
Solutions of homework (in written form, better if as LaTeX-PDF file) are necessary to be eligible to the exam.
They can be elaborated in team, however each student wishing take the exam should send me her/his proper
copy of solutions; the use of any textbook is allowed (and strongly recommended), but must be declared.
Moreover, the exam will be:
- in oral form, for peoples attending regularly the lessons,
- a written exam (3/4 problem, 2 hours) + oral discussion, for peoples who do not attend the lessons.
Homeworks for the current A.Y.
| Homework 1 (2023) PDF, TeX | Homework 2 (2023) PDF, TeX | Homework 3 (2023) PDF, TeX |
Homeworks for the passed years:
| Homework 1 (2021) PDF, TeX | Homework 2 (2021) PDF, TeX | Homework 3 (2021) PDF, TeX |
| Homework 1 (2019) PDF, TeX | Homework 2 (2019) PDF, TeX | Homework 3 (2019) PDF, TeX |
| Homework 1 (2017) PDF, TeX | Homework 2 (2017) PDF, TeX | Homework 3 (2017) PDF, TeX |
| Homework 1 (2016) PDF, TeX | Homework 2 (2016) PDF, TeX | Homework 3 (2016) PDF, TeX |
| Homework 1 (2015) PDF, TeX | Homework 2 (2015) PDF, TeX | Homework 3 (2015) PDF, TeX |
| Homework 1 (2014) PDF, TeX | Homework 2 (2014) PDF, TeX | Homework 3 (2014) PDF, TeX |
| Homeworks (2012) PDF, TeX |
Exam sessions
- to be scheduled
This work is licensed under a Creative Commons License.
Responsabile del sito: Giuseppe Molteni
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