Skip to main content

MA4L6 Content

Content:
The course will cover some of the following topics, depending on time and audience preferences:

  • Warm-up:
    The counting functions $\pi(x), \Psi(x)$ of primes up to $x$. Chebychev's upper and lower bounds for $\Psi(x)$.
  • Basic theory of the Riemann zeta function:
    Definition of the zeta function $\zeta(s)$ when $\Re(s) > 1$, and then when $\Re(s) > 0$ and for all $s$. The connection with primes via the Euler product. Proof that $\zeta(s) \neq 0$ when $\Re(s) \geq 1$, and deduction of the Prime Number Theorem (asymptotic for $\Psi(x)$).
  • More on zeros of zeta:
    Non-existence of zeta zeros follows from estimates for $\sum_{N < n < 2N} n^{it}$. The connection with exponential sums, and outline of the methods of Van der Corput and Vinogradov. Wider zero-free regions for $\zeta(s)$, and application to improving the Prime Number Theorem. Statement of the Riemann Hypothesis.
  • Primes in arithmetic progressions:
    Dirichlet characters $\chi$ and Dirichlet $L$-functions $L(s,\chi)$. Non-vanishing of $L(1,\chi)$. Outline of the extension of the Prime Number Theorem to arithmetic progressions.

Aims:
Multiplicative number theory studies the distribution of objects, like prime numbers or numbers with ''few'' prime factors or ''small'' prime factors, that are multiplicatively defined. A powerful tool for this is the analysis of generating functions like the Riemann zeta function $\zeta(s)$, a method introduced in the 19th century that allowed the resolution of problems dating back to the ancient Greeks. This course will introduce some of these questions and methods.

Objectives:
By the end of the module the student should be able to:

  • Consolidate existing knowledge from real and complex analysis and be able to place in the context of Analytic Number Theory
  • Have a good understanding of the Riemann zeta function and the theory surrounding it up to the Prime Number Theorem
  • Understand and appreciate the connection of the zeros of the zeta function with exponential sums and the statement of the Riemann Hypothesis
  • Demonstrate the necessary grasp and understanding of the material to potentially pursue further postgraduate study in the area

Books:

  • H. Davenport. Multiplicative Number Theory. Third edition, published by Springer Graduate Texts in Mathematics. 2000
  • A. Ivi'c. The Riemann Zeta-Function. Theory and Applications. Dover edition, published by Dover Publications, Inc.. 2003
  • H. Montgomery and R. Vaughan. Multiplicative Number Theory I. Classical Theory. Published by Cambridge studies in advanced mathematics. 2007
  • E. C. Titchmarsh. The Theory of the Riemann Zeta-function. Second edition, revised by D. R. Heath-Brown, published by Oxford University Press. 1986