## 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 or MA3RCA Real and Complex Analysis and MA3Z7 Number Theory
Non-modular pre-requisites: Part 3 students must take MA3Z7 Number Theory as a co-requisite. MA4Z7 can be taken as a co-requisite in place of MA3Z7
Co-requisites:
Modules excluded:
Current from: 2020/1

Module Convenor: Dr Titus Hilberdink

Type of module:

Summary module description:

This module introduces complex analytic techniques in the theory of numbers.

Aims:

To introduce complex analytic techniques in the theory of  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.

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. We make a study of the Gamma function. We develop the theory of Dirichlet series of a complex variable and study 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:
 Autumn Spring Summer Lectures 20 Guided independent study: 80 Total hours by term 0 0 Total hours for module 100

Summative Assessment Methods:
 Method Percentage Written exam 100

Summative assessment- Examinations:
2 hours.

Summative assessment- Coursework and in-class tests:

Formative assessment methods:
Problem sheets.

Penalties for late submission:

The Module Convenor 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[1] (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: http://www.reading.ac.uk/web/FILES/qualitysupport/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.