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 BombieriWirsing's ``elementary'' argument (giving also a good estimate for the remainder term).

Topic 2: 
Some results about sieve methods. The Selberg Lambdasquare method, BrunTitchmarsh'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 ErdősTuràn 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 
10.3012.30 
Room 5 
Friday 
11.3013.30 
Room 5 
First lesson: Monday 6, March 2017!
 Notes for the course (work in progress...)
 The draft copy of the original paper of Riemann in Number Theory
Exam
Solutions of homeworks (in written form, better if as LaTeXPDF 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.
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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