Analytic Number Theory - 2023 entry

The study of properties of the integer numbers, in particular of prime numbers, is one of the most ancient topics in mathematics. The study of the prime numbers and their distribution is also considered to be one of the most beautiful topics in mathematics. Analytic number theory is the area of mathematics that uses methods from mathematical analysis to solve problems about the integers. After reviewing some basic concepts in elementary number theory, the main aim of this lecture course is to show how powerful mathematical analysis is, in particular, complex analysis, in the study of the distribution of prime numbers.

Pre-requisite Module: MTH2001 or MTH2009 or equivalent. 

The aim of this course is to introduce you to the theory of prime numbers, showing how the irregularities in this sequence of integer numbers can be tamed by the power of complex analysis. The main aim of the course is to present a proof of the Prime Number Theorem which is the corner-stone of prime number theory. We also will discuss the Riemann Hypothesis, which is arguably the most important unsolved problem in modern mathematics.

From this module, you will acquire a working knowledge of the main concepts of analytic number theory, together with some appreciation of modern results and techniques, and some recent research in the area.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Recall key definitions, theorems and proofs in analytic number theory;

2 Apply the techniques of complex analysis and analytic number theory to solve a range of seen and unseen problems;

3 Discuss some aspects of modern research in analytic number theory;

4 Explore the distribution of prime numbers using complex analysis;

Discipline Specific Skills and Knowledge:

5 Explain the relationship between the topics in this module and other material in number theory, complex analysis and cognate areas of pure mathematics taught elsewhere on the programme;

6 Apply a range of techniques from the module with precision and clarity;

Personal and Key Transferable / Employment Skills and Knowledge:

7 Show enhanced problem-solving skills and ability to formulate your solutions as mathematical proofs;

8 Communicate your work professionally, and using correct mathematical notation.

1 Fundamental Theorem of Arithmetic and Some Foundations on Prime Numbers:

- Divisibility;

- Prime Numbers;

- The fundamental theorem of arithmetic;

- The Euclidean algorithm;

2 Arithmetic Functions:

- The Mobius function;

- The Euler function;

- The Dirichlet convolution;

- The Mangoldt function;

- Multiplicative functions;

3 Averages of Arithmetic Functions:

- The big oh notation;

- Euler's summation formula;

- Some elementary asymptotic formulas;

- Average order of some other arithmetic functions:

4 Elementary Theorems of the Distribution of Prime Numbers:

- Chebyshev's functions:

- Equivalent forms of the prime number theorem;

- Shapiro's Tauberian theorem;

- Sums over primes;

5 Dirichlet Characters, Dirichlet Series and Euler Products:

- Definitions;

- The character group;

- Dirichlet characters;

- Dirichlet series;

- The non-vanishing of L-functions:

6 Dirichlet’s Theorem on Primes in Arithmetic Progression:

- Dirichlet's theorem for primes of the form 4n-1;

- The plan of the proof of Dirichlet's theorem;

- Distribution of primes in arithmetic progressions;

7 Riemann zeta function:

- Definitions;

- Analytic continuation;

- Functional equation;

- Non-vanishing on Re(s) = 1;

8 Proof of the Prime Number Theorem;

9 The Riemann Hypothesis and its significance;

*Please note: some of the topics above are subject to change.

Reassessment will be by written exam in the deferred, or failed, element only. For deferred candidates, the module mark will be uncapped. For referred candidates the module mark will be capped at 50%.

Basic Reading:

ELE: http://vle.exeter.ac.uk/

Reading list for this module: