Topics

This course is centered around the breakthrough paper Multiplicative functions in short intervals  by Matomäki and Radziwiłł, which relates short averages of a multiplicative function to long averages. In particular, this implies that the Möbius function has mean value 0 over almost all short intervals [x,x+h] for any h=h(x) which grows with x.

The main goal of the course is to develop the required tools and prove their main result. The tools that the student will learn during the course have many other applications, for instance, to prime numbers in short intervals.

The work of Matomäki and Radziwiłł has seen spectacular applications in recent years, including Tao's proof of the logarithmic Chowla conjecture and solution of the Erdös discrepancy problem 

Topics covered include

• Dirichlet polynomial techniques

• Mellin transform

• Anatomy of integers

• Multiplicative functions

Prerequisites

It is recommended, but not strictly required, that the student has seen a proof of the Prime number theorem, and has some familiarity with the tools that go with it (zeta function, Perron formula, etc.). All the relevant tools will be developed during the course

Lectures

Mondays 10am to 12noon, starting  22nd April. Lecture room VC1 in Oxford Mathematical Institute, online on Teams

Please note: no lectures on Bank Holidays (6 and 27 May).  Replacement lectures will take place 09.00 - 11.00 on Friday 10 May, and 09.00 - 11.00 on Friday 31 May). 

Lecture notes

Lecture notes:  MRnotes 

Notes will be updated along the course

Credit

If you wish to get credit for the course, let me know. This can be done with a small written assignment, for instance, on an application of the topics from the course.

Registration

TCC courses webpage , email to: tcc@maths.ox.ac.uk with your MS Teams email address