Dr Davoud Cheraghi


Davoud Cheraghi


Math M3/4/5 P16: Analytic Number Theory

Instructor: Davoud Cheraghi
E-mail: d.cheraghi@imperial.ac.uk
Office: Huxley 683
Office hours: Tuesdays 17:00-18:00, and Thursdays 12:00-13:00
Student representative for the course: Mr Petru Constantinescu (PC4113@ic.ac.uk)

Prerequisites

It will be assumed that students have had a previous course in number theory (preferably M3P14, or a course similar to that). Also, the students must have had the usual undergraduate training in analysis and a strong course in complex analysis (preferably M2PM3, or a similar course to the level of the residue theorem), although the complex analysis course could be taken concurrently.

Lectures
  • Tuesdays (Jan 12 to March 22), 14:00-16:00, in Room Huxley 140
  • Thursdays (Jan 14 to March 17), 14:00-15:00, in Room Huxley 140
The classes on the following dates will be problem solving sessions: Feb 4,18, and Mar 3, 17, 22.

Topics we cover in the course
  • Introductory material on primes
  • Arithmetic functions: Möbius function, Euler function, Divisor function, Sigma function
  • Dirichlet series, Euler products, von Mangoldt function
  • Riemann Zeta-function, analytic continuation to Real(s) > 0
  • Non-vanishing of Zeta(s) on Real(s)=1
  • Proof of the prime number theorem
  • The Riemann hypothesis and its significance
  • The Gamma function, the functional equation for Zeta(s), the value of Zeta(s) at negative integers

Lecture notes

You may download the lecture notes by clicking here.

Homeworks

Problem Sheet 1, Problem Sheet 2, Problem Sheet 3, Problem Sheet 4.

Solutions to PS 1, Solutions to PS 2, Solutions to PS 3, Solutions to PS 4.

Assessment

There will be two progress tests that will form 10 percent of the total mark for the course. These will be on Feb 18 and March 17.
Test 1, Test 2, Solutions to Test 1, Solutions to Test 2.

The remaining 90 percent is determined by a two-hour written examination. This will be on 25 May 2016, from 14:00 to 16:00 in Huxley building Rooms 340, 341, and 342.

Here are the final exam, and the solutions. I am very happy with how most of you did at the final.


References for further reading
  • T. M. Apostol, Introduction to Analytic Number Theory
  • H. Davenport, Multiplicative Number Theory
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers
  • I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers