Advanced Topics in Analytic Number Theory
A.Y. 2024/2025
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.
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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Topic 1: Irrationality: some easy results and some not so easy classical results about special numbers: e, pi and others.
Topic 2: Continued fractions.
Topic 3: Lindemann-Weierstrass theorem, six exponential theorem, Lang-Schneider theorem, Gelfond-Schneider theorem and transcendence of the main invariants for complex elliptic curves.
Topic 4: Baker's results about sums of logarithms and some of its consequences, in particular the solution of Gauss' problem about the classification of quadratic fields with unique factorization.
Topic 2: Continued fractions.
Topic 3: Lindemann-Weierstrass theorem, six exponential theorem, Lang-Schneider theorem, Gelfond-Schneider theorem and transcendence of the main invariants for complex elliptic curves.
Topic 4: Baker's results about sums of logarithms and some of its consequences, in particular the solution of Gauss' problem about the classification of quadratic fields with unique factorization.
Prerequisites for admission
Basic notions of Complex Analysis and of the theory of rings in number fields. The course named "Analytic Number Theory" is NOT required.
Teaching methods
Lectures by the teacher. Attendance at classes, although not mandatory, is strongly recommended.
Teaching Resources
G. Molteni: Written notes (available day by day).
M. Ram Murty, Purusottam Rath: Transcendental Numbers, Springer, New-York, 2014.
J. Borwein, A. van der Poorten, J. Shallit, W. Zudilin: Neverending fractions, Cambridge University Press, Cambridge, 2014.
A. M. Rockett, P. Szusz: Continued fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.
M. Ram Murty, Purusottam Rath: Transcendental Numbers, Springer, New-York, 2014.
J. Borwein, A. van der Poorten, J. Shallit, W. Zudilin: Neverending fractions, Cambridge University Press, Cambridge, 2014.
A. M. Rockett, P. Szusz: Continued fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992.
Assessment methods and Criteria
Oral exam about the topics of the course.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor:
Molteni Giuseppe
Shifts:
Turno
Professor:
Molteni GiuseppeEducational website(s)
Professor(s)
Reception:
My office: Dipartimento di Matematica, via Saldini 50, first floor, Room 1044.