Module Information | Study information

MODULE TITLE	Topics in Analytic Number Theory	CREDIT VALUE	15
MODULE CODE	MTHM061	MODULE CONVENER	Dr Julio Andrade (Coordinator)

DURATION: TERM	1	2	3
DURATION: WEEKS	12

Number of Students Taking Module (anticipated)	22

DESCRIPTION - summary of the module content

This is an advanced 4^th-year module in pure mathematics. In this module, we will be studying some advanced topics in number theory that are drawn from recent research in the area and/or from topics not normally covered in the undergraduate degree.

Some possible topics to be covered include the theory of the Riemann zeta-function, sieve methods, elliptic curves, and random matrix theory. (For a more comprehensive list of topics see Syllabus Plan below).

The focus of the module will be on investigating and discussing some basic problems related to the chosen topics and studying some techniques and applications. This module is suitable for students who have interests in number theory and pure mathematics in general. This module is also suitable if you are thinking about going forward and work with research in mathematics and/or doing a PhD in mathematics

Another novelty of this module is that it might be taught by using a mix of methods including (but not restricted to): traditional lectures, student seminars and presentations, reading assignments, and Moore’s method, which is a deductive manner of instruction used in advanced mathematics course where the content of the course is presented in whole or in part by the students themselves.

Pre-requisite Module: MTH2009 and MTH3004

AIMS - intentions of the module

This course aims to introduce to the student some recent and/or advanced topics in number theory. The topics will be chosen from either recent research in the area and/or from advanced topics not normally seen in an undergraduate course. The aim is to discuss some problems, techniques, and applications of studying certain topics in analytic number theory and auxiliary areas. We aim to present fundamental results with detailed proofs and at the same time highlight the tight connection between several subjects in mathematics, especially analytic number theory, elementary number theory, complex analysis, and analysis in general.

From this module, the student should be able to acquire a working knowledge of some of the main concepts appearing in some areas of analytic number theory and correlated areas, together with some appreciation of modern results and techniques. Various applications will be highlighted.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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 and correlated areas
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 and correlated areas;

Discipline Specific Skills and Knowledge

4. 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;
5. Apply a range of techniques from the module with precision and clarity;

Personal and Key Transferable / Employment Skills and Knowledge

6. Show enhanced problem-solving skills and ability to formulate your solutions as mathematical proofs
7. Communicate your work professionally, and using correct mathematical notation

8. Illustrate oral presentation skills by participation in seminars and oral presentation

SYLLABUS PLAN - summary of the structure and academic content of the module

A selection of possible topics to be covered in this module is shown below. Notice that the topics may change from year to year, and they might include (but not restricted to):

In any year a selection of at least two topics will be taken from the following list:

Basics in Analytic Number Theory and the Prime Number Theorem.
The Theory of the Riemann zeta-function and other L-functions.
Zero-free regions, moments of L-functions and distribution of zeros of L-functions.
Random Matrix Theory, Quantum Chaos and Applications to Number Theory
Sieve Methods.
Hardy-Littlewood Circle Method.
Diophantine Equations/Analysis.
Some topics in Probabilistic Number Theory.
Automorphic Forms and Spectral Theory of Automorphic Forms.
Elliptic Curves.
Number Theory in Function Fields.
Arithmetic of Quadratic Forms.
Modular Forms.
Topics in Additive Number Theory.
Elliptic Functions.
Partitions.
Exponential Sums.
Other possible interesting/recent topics in number theory and/or related areas.

Notice that in the first lecture for this module, the lecturer will clearly state what are the topics that are planned to be covered in that year.

LEARNING AND TEACHING

LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)

Scheduled Learning & Teaching Activities	48	Guided Independent Study	102	Placement / Study Abroad	0

DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS

Category	Hours of study time	Description
Scheduled Learning and Teaching Activities	48	Lectures
Guided Independent Study	102	Assessment preparation; private study, problem sheets, office hours, and possible tutorials.

ASSESSMENT

FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade

Form of Assessment	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
Problem Sheets	Standard problem sheet for a 4^th year pure mathematics module.	1, 2, 3, 4, 5, 6, 7	Written and verbal.
Reading Assignments	Chapters of lecture notes, books and/or papers to be read before the lectures.	1, 2, 3, 4, 5, 6, 7	Verbal.

SUMMATIVE ASSESSMENT (% of credit)

Coursework	40	Written Exams	40	Practical Exams	20

DETAILS OF SUMMATIVE ASSESSMENT

Form of Assessment	% of Credit	Size of Assessment (e.g. duration/length)	ILOs Assessed	Feedback Method
Coursework 1	10	Problem Sheet	1, 2, 3, 4, 5, 6, 7	Written comments
Coursework 2	10	Problem Sheet	1, 2, 3, 4, 5, 6, 7	Written comments
Coursework 3: Written Observation/Critic	10	Observation/critic of a lecture and/or presentation	3, 4, 6, 7,8	Written comments
Coursework 4: Written Task	10	Written notes and/or slides of the lecture/presentation.	3, 4, 5, 6, 7,8	Written comments
Presentations/Seminars	20	It will take place during the teaching term in the lectures.	1, 2, 3, 4, 5, 6, 7,8	Written and/or verbal
Written Exam	40	Written examination – 2 hours (Summer)	1, 2, 3, 4, 5, 6, 7	Feedback sheet (usual feedback in the script)

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)

Original Form of Assessment	Form of Re-assessment	ILOs Re-assessed	Time Scale for Re-assessment
Coursework 1*	Coursework 1	1, 2, 3, 4, 5, 6, 7	Referral/defderral period
Coursework 2*	Coursework 2	1, 2, 3, 4, 5, 6, 7	Referral/defderral period
Coursework 3*	Coursework 3	3,4,6,7,8	Referral/defderral period
Coursework 4*	Coursework 4	3,4,5,6,7,8	Referral/defderral period
Presentations/Seminars	Presentation	1, 2, 3, 4, 5, 6, 7, 8	Referral/defderral period
Written Exam	Written exam	1, 2, 3, 4, 5, 6, 7	Referral/defderral period

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

RESOURCES

INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

Basic reading:

Author	Title	Publisher	Year
Apostol, T.	Introduction to Analytic Number Theory	Springer-Verlag	1976
Davenport, H.	Multiplicative Number Theory	Springer-Verlag	2000
Hardy, G.H. and Wright, E.M.	An Introduction to the Theory of Numbers	Oxford University Press	2008
Jameson, G.J.O.	The Prime Number Theorem	Cambridge University Press	2003
Montgomery, H.L. and Vaughan, R.C.	Multiplicative Number Theory, I: Classical Theory	Cambridge University Press	2007
Stopple, J.	A Primer of Analytic Number Theory	Cambridge University press	2003
Vaughan, R.C.	The Hardy-Littlewood Method	Cambridge University Press	1997
Rassias, M. Th.	Goldbach’s Problem: Selected Topics	Springer-Verlag	2017
Steuding, J. and et.al.	Diophantine Analysis: Course Notes from a Summer School	Springer-Verlag	2016
Hindry, M. and Silverman, J.H.	Diophantine Geometry: An Introduction	Springer-Verlag	2000
Cohen, H.	Number Theory: Volume I: Tools and Diophantine Equations	Springer-Verlag	2007
Andreescu, T. and Andrica, D.	Quadratic Diophantine Equations	Springer-Verlag	2015
Andreescu, T. and Andrica, D. et.al.	An Introduction to Diophantine Equations: A problem-based approach	Springer-Verlag	2010
Rosen, M.	Number Theory in Function Fields	Springer-Verlag	2002
Titchmarsh, E.C.	The Theory of the Riemann Zeta-function	Oxford University Press	1987
Iwaniec H. and Kowalski, E.	Analytic Number Theory	American Mathematical Society	2004
Cojocaru, A.C. and Murty, M.R.	An Introduction to Sieve Methods and Their Applications	Cambridge University Press	2005
Tenenbaum, G.	Introduction to Analytic and Probabilistic Number Theory	Cambridge University Press	1995
Lang, S.	Introduction to Modular Forms	Springer-Verlag	1976
Koblitz, N.	Introduction to Elliptic Curves and Modular Forms	Springer-Verlag	1993
Iwaniec, H.	Spectral Methods of Automorphic Forms	American Mathematical Society	2002
Shimura, G.	Arithmetic of Quadratic Forms	Springer-Verlag	2010
Gerstein, L.J.	Basic Quadratic Forms	American Mathematical Society	2008
Andrews, G.	The Theory of Partitions	Cambridge University Press	1998
Andrews, G. and Eriksson, K.	Integer Partitions	Cambridge University Press	2004
Lang, S.	Elliptic Functions	Springer-Verlag	1987
Nathanson, M. B.	Additive Number Theory: The Classical Bases	Springer-Verlag	1996
Tao, T. and Vu, V.H.	Additive Combinatorics	Cambridge University Press	2010
Graham, S.W. and Kolesnik, G.	Van der Corput’s Method of Exponential Sums	Cambridge University Press	1991
Korobov, N.M.	Exponential Sums and Their Applications	Springer-Verlag	1992
Gaitsgory, D. and Lurie, J.	Weil’s Conjecture for Function Fields	Princeton University Press	2019
Bernstein, J. and Gelbart, S.	An Introduction to the Langlands Program	Birkhauser	2003

Reading list for this module:

There are currently no reading list entries found for this module.

CREDIT VALUE	15	ECTS VALUE	7.5

PRE-REQUISITE MODULES	None
CO-REQUISITE MODULES	None

NQF LEVEL (FHEQ)	7	AVAILABLE AS DISTANCE LEARNING	No
ORIGIN DATE	Monday 11th March 2024	LAST REVISION DATE	Tuesday 19th March 2024

KEY WORDS SEARCH	Analytic number theory, number theory in function fields, Riemann zeta-function, L-functions, circle-method, random matrix theory, function fields, sieve methods, automorphic forms, modular forms.

Topics in Analytic Number Theory - 2024 entry