# Analytic Number Theory

## Unit aims

To gain an understanding and appreciation of analytic number theory, and some of its most important achievements. To be able to apply the techniques of complex analysis to study a range of specific problems in number theory.

## Unit description

The study of prime numbers is one of the most ancient and beautiful topics in mathematics. After reviewing some basic results in elementary number theory and the theory of Dirichlet characters and L-functions, the main aim of this lecture course will be to show how the power of complex analysis can be used to shed light on irregularities in the sequence of primes. Significant attention will be paid to developing the theory of the Riemann zeta function. The course will build up to a proof of the Prime Number Theorem and a description of the Riemann Hypothesis, arguably the most important unsolved problem in modern mathematics.

## Relation to other units

This is one of three Level 6 and Level 7 units which develop number theory in various directions. The others are Number Theory and Algebraic Number Theory.

## Learning objectives

To gain an understanding and appreciation of Analytic Number Theory and some of its important applications. To be able to use the theory in specific examples.

## Transferable skills

Using an abstract framework to better understand how to attack a concrete problem.

## Syllabus

Preliminary review. Elementary results on the distribution of primes. Equivalent forms of the Prime Number Theorem. Chebychev's order of magnitude result for the prime counting function.

Definition and basic properties of arithmetic functions. Dirichlet convolution and the ring of arithmetic functions. Mobius inversion.

Summation techniques and average orders of arithmetic functions. Partial summation.

Review of complex analysis. Dirichlet series and Euler products. Perron's formula.

Analytic properties of the Riemann zeta function. The gamma function and the functional equation for the zeta function. The Riemann Hypothesis.

Size of the zeta function in the critical strip. Non-vanishing of the zeta function on the line s=1. Analytic proof of the prime number theorem. Equivalent forms of the Riemann Hypothesis.

Dirichlet characters and Dirichlet L-functions. Dirichlet's theorem on primes in arithmetic progression.

Gauss sums. The Polya-Vinogradov inequality.

Other possible topics as time permits: The class number formula, more about primes in arithmetic progressions, sieve techniques, additive number theory.

## Reading and References

1. T. M. Apostol, Introduction to analytic number theory. Springer, 1976.
2. H. Davenport, Multiplicative Number Theory, third edition, Springer 2000
3. H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press 2007
4. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995.

Unit code: MATHM0007
Level of study: M/7
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturers: Dr Min Lee and Jos Gunns

## Pre-requisites

MATH33000 Complex Function Theory

None

## Methods of teaching

Lectures and exercises.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

• 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.