Advanced Topics in Analytic Number Theory

A.Y. 2020/2021
6
Max ECTS
42
Overall hours
SSD
MAT/05
Language
English
Learning objectives
A clever but now very easy argument due to Cantor proves that almost every real number is a
transcendental number. Nevertheless, proving that a given number is transcendental or even only
irrational is a much more complicated task needing stronger and stronger methods of proof. The course
some of the main topics about this problem will be discussed.
Expected learning outcomes
Students will know the basic results about irrationality and transcendence of numbers and some of fundamental methods for their proof.
Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
The course is scheduled for the second semester of the academic year 2020-2021. Moreover, the number of students expected is in any case compatible with a frontal teaching even in the presence of social distancing rules. For this reason, I'm confident that the course will be delivered in its usual format, i.i., classroom both for teaching and for exercises.
If this is not possible, the course will be delivered through recorded lessons plus PDF files provided through the Ariel university platform.
Course syllabus
Irrationality: some easy results. Famous transcendendal numbers: e, π and others. Continued fractions and Padé approximations. Baker's results about sums of logarithms. Subspace theorem.
Prerequisites for admission
The contents in analysis 1-2-3 and basic parts of Complex Analysis. Despite its name, the course wants to result accessible also by students without a specific training in number theory, in particular only an elementary knowledge of arithmetic number theory is strictly necessary.
Teaching methods
Traditional way for this type of courses (i.e. lessons in a classroom).
Teaching Resources
Notes from the teacher, plus selected chapters from different books, for example:

M. Ram Murty, Purusottam Rath: Transcendental Numbers, Springer, New-York, 2014.

M. Waldschmidt: Nombres Transcendants, Lectures notes in Mathematics 402, Springer, 1974.
Assessment methods and Criteria
In order to pass this exam the student has to produce written (and correct, of course) solutions of the Homeworks I will propose during the course, plus a final oral exam about the main topics of the course.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Molteni Giuseppe
Professor(s)
Reception:
My office: Dipartimento di Matematica, via Saldini 50, first floor