MA4ANT-Analytic Number Theory

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites: MA2RCA Real and Complex Analysis and MA3Z7 Number Theory
Non-modular pre-requisites: MA4Z7 can be taken as a co-requisite in place of MA3Z7
Co-requisites:
Modules excluded: MA3ANT Analytic Number Theory
Current from: 2023/4

Module Convenor: Dr Jani Virtanen
Email: j.a.virtanen@reading.ac.uk

Type of module:

Summary module description:

This module introduces complex analytic techniques in the theory ofÌýnumbers.

Aims:

To introduce complex analytic techniques in the theory ofÌý numbers.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

• use real analytic methods in the theory of arithmetical functions; find analytic continuations of functions;


• formulate and solve problems using Dirichlet series;


• appreciate the proof of the prime number theorem and the link with the Riemann zeta function.

This module will be assessed to a greater depth than the excluded module MA3ANT.

Additional outcomes:

Outline content:

This module uses techniques from complex analysis to study problems in number theory. First we develop some necessary real and complex analysis, in particular the notion of analytic continuation. After studying the Gamma function, we develop the theory of Dirichlet series of a complex variable. This will be followed by a study of the Riemann zeta function – a central object in analytic number theory – and its connection to the distribution of prime numbers.

Brief description of teaching and learning methods:
Lectures supported by problem sheets

Contact hours:

Ìý	Autumn	Spring	Summer
Lectures		20
Guided independent study:		80
Ìý	Ìý	Ìý	Ìý
Total hours by term	0		0
Ìý	Ìý	Ìý	Ìý
Total hours for module	100

Summative Assessment Methods:

Method	Percentage
Written exam	100

Summative assessment- Examinations:

2 hours.

The examination for this module will require a narrowly defined time window and is likely to be held in a dedicated exam venue.

Summative assessment- Coursework and in-class tests:

Formative assessment methods:
Problem sheets.

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/cqsd-old-site-documents/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:
A mark of 50% overall.

Reassessment arrangements:
One examination paper of 2 hours duration in August/September.

Additional Costs (specified where applicable):

1) Required text books:Ìý

2) Specialist equipment or materials:Ìý

3) Specialist clothing, footwear or headgear:Ìý

4) Printing and binding:Ìý

5) Computers and devices with a particular specification:Ìý

6) Travel, accommodation and subsistence:Ìý

Last updated: 30 March 2023