Lecturer: Adam Harper

Term(s): Term 1

Commitment: 30 one hour lectures

Assessment: Oral Exam

Formal registration prerequisites: None

Assumed knowledge: Some basic real and complex analysis, including: uniform convergence, the Identity Theorem from complex analysis and especially Cauchy's Residue Theorem. This is covered in the modules MA244 Analysis III and (ideally) MA3B8 Complex Analysis. Although the module will not assume much specific content or results, it will have a serious "analytic" flavour of estimating objects and handling error terms. The most important thing is to be comfortable with this style of mathematics, which might be familiar from previous courses in analysis, measure theory or probability.

Useful background: The module will assume very little from number theory, but in a few places it will be useful to know things like: the Chinese Remainder Theorem, the structure of the multiplicative group mod q. These are covered in e.g. MA249 Algebra II and MA257 Introduction to Number Theory.

Synergies: This is fundamentally an analysis module, and would go well (in terms of the general style of the mathematics) with modules like: MA633 Fourier Analysis, MA6J0 Advanced Real Analysis and possibly MA627 Ergodic Theory. Those interested in number theory will probably also enjoy MA626 Elliptic Curves.

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

Additional Resources