Study information

# Analytic Number Theory - 2023 entry

MODULE TITLE CREDIT VALUE Analytic Number Theory 15 MTHM041 Dr Julio Andrade (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 30
DESCRIPTION - summary of the module content

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.

AIMS - intentions of the module

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.

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;

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.

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

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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures Guided Independent Study 117 Assessment preparation; private study

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
Coursework - Problem Sheets 10 hours per sheet (3 or 4 sheets) All Written and Verbal

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Exam – Closed Book 100 2 hours (Summer) All Exam mark (results released online); individual feedback, upon request.

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
All Above Written Examination (100%) All August Ref/Def Period

RE-ASSESSMENT NOTES

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%.

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

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Apostol, T. Introduction to Analytic Number Theory Undergraduate Texts in Mathematics, Springer-Verlag 1976
Set Davenport, H. Multiplicative Number Theory Springer-Verlag, Graduate Texts in Mathematics 2000
Set Hardy, G.H. and Wright, E.M. An Introduction to the Theory of Numbers Oxford University Press 2008
Set Jameson, G.J.O. The Prime Number Theorem LMS Student Texts, (Vol. 53) Cambridge 2003
Set Montgomery, H.L. and Vaughan, R.C. Multiplicative Number Theory, I: Classical Theory Cambridge Advanced Maths, Vol 97 Cambridge University Press 2007
Set Stopple, J. A Primer of Analytic Number Theory Cambridge 2003
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2001, MTH2009
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Tuesday 10th July 2018 Thursday 26th January 2023
KEY WORDS SEARCH Analytic Number Theory; Prime Number Theorem; The Riemann Zeta-Function

Please note that all modules are subject to change, please get in touch if you have any questions about this module.