Analytic Number Theory (topics) 2016/'17

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 real analysis; some familiarity with the complex analysis will be also useful. The course touches four main topics of the Analytic Number Theory:

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
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

Monday 10.30-12.30 Room 5
Friday 11.30-13.30 Room 5

First lesson: Monday 6, March 2017!


Exam

Solutions of homeworks (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:

Homeworks for the current A.Y.

Homework 1 (2017) PDF, TeX Homework 2 (2017) PDF, TeX Homework 3 (2017) PDF, TeX

Homeworks for the passed years:

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 Homework 2 (2014) PDF Homework 3 (2014) PDF
Homeworks (2012) PDF


Exam sessions



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