Lecturer

Özlem Imamoglu

Coordinator

Emilio Corso

Room

HG G 19.1, except for 23/09 and 11/11, in HG G 19.2

Time

Thursday, 12.00-14.00

Content

The purpose of this seminar is to introduce a diversified range of topics in elementary number theory, each of which has spurred, and still motivates, a great deal of research in the area. This will hopefully encourage a deeper understanding of the subject and serve as a basis for more advanced courses.

Prerequisites

Algebra I and II, the basics of complex analysis.

Schedule

Each student is expected to lecture on one of the subjects listed below. For each meeting, there will be two students presenting the assigned topic. On the first week, we will distribute the subjects among the participants. The talks will start on the third week.

Date	Talk	Speakers	References	Notes
23rd September 2021	Overview and distribution of the talks
30th September 2021	No class
7th October 2021	Quadratic residues and the representation of a number as a sum of four squares	Jean-Luc Portner, Matthias Schmickler	[C] Chapter IV, [HW] Section 20.11, [FB] Chapter VI.4 and VII.1	Quadratic residues and the representation of a number as a sum of four squares
14th October 2021	The law of quadratic reciprocity	Loic Pierre Dobler, Paul Mangers Bastian	[C] Chapter 5, [A] Chapter 9	The law of quadratic reciprocity
21st October 2021	Rational approximation of irrationals and Hurwitz's theorem	Lukas Dundulis, Xinmeng Zeng	[C] Chapter 3, [F]	Rational approximation of irrationals and Hurwitz's theorem
28th October 2021	Introduction to transcendental numbers	Caroline Braun, Romeo Schilling	[B] Chapter 1, [S] pages 1-7, [MR] Chapters 5-6-7-8	Introduction to transcendental numbers, The Lindemann-Weierstrass theorem
4th November 2021	Arithmetical functions and Dirichlet multiplication	Alexander Basler, Anselm Schönbächler	[A] Chapter 2 Sections 2.1-2.12, [HW] Chapter XVI	Arithmetical functions and Dirichlet multiplication
11th November 2021	Average orders of arithmetical functions	Patrick Edera, Nina Goldhirsch	[A] Chapter 3 Sections 3.1-3.9, [C] Chapter VI, [HW] Chapter XVIII	Average orders of arithmetical functions
18th November 2021	Generating functions of arithmetical functions	Jana Dinger, Pablo Lahmann	[HW] Chapter XVII	Generating functions of arithmetical functions
25th November 2021	Chebyshev's theorem on the distribution of prime numbers	Nikolaos Stamatopoulos, Zhiang Wu	[C] Chapter VII, [A] Chapter 4	Chebyshev's theorem on the distribution of prime numbers
2nd December 2021	Weyl's theorems on uniform distribution and Kronecker's theorem	Philippe Golliard, Fabian Richner	[C] Chapter VIII, [HW] Chapter XXIII	Weyl's theorems on uniform distribution and Kronecker's theorem
9nd December 2021	Partitions	Helena Bösch, Pia Herkenrath	[A] Chapter 14, [HW] Chapter XIX Sections 19.1-19.8	Partitions
16th December 2021	Reduction of binary quadratic forms	Sina Keller, Florian Trummer	[Bu] Chapters 1, 2 and 3	Reduction of binary quadratic forms

Literature

• [HR] An introduction to the theory of numbers (5th edition) by G.H. Hardy and E.M. Wright (Oxford University Press 1980)
• [A] Introduction to Analytic Number Theory by T.M. Apostol (Springer 1976)
• [C] Introduction to Analytic Number Theory by K. Chandrasekharan (Springer 1968)
• [FB] Complex Analysis by E. Freitag and R. Busam (Springer 2009)
• [MR] Transcendental Numbers by M.R. Murty and P. Rath (Springer 2014)
• [B] Transcendental Number Theory by A. Baker (Cambridge University Press 1975)
• [Bu] Binary Quadratic Forms, Classical Theory and Modern Computations by D.A. Buell (Springer-Verlag 1989)
• [S] Transcendental Number Theory by K. Soundararajan (Course notes), available here
• [F] A Geometrical Proof of a Theorem of Hurwitz by L.R. Ford (Proceedings of the Edinburgh Mathematical Society 35 (1916) 59-65)