# Mathematics

## ECMM741 - Analytic Number Theory (2018)

MODULE TITLE CREDIT VALUE Analytic Number Theory 15 ECMM741 Dr Julio Andrade (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 Number of Students Taking Module (anticipated) 16
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.

Prerequisite module: ECM3703

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 theorme

- 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

*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
Not applicable.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 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 80 2 hours All Exam mark. Results released online. Individual feedback upon request
Coursework - problem sheets 20 Variable All Written and verbal

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

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

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

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

Other Resources: