# Analytic Number Theory - MAT00051M

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• Department: Mathematics
• Module co-ordinator: Prof. Sanju Velani
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2018-19

## Module summary

Pre-requisite Module(s)

Complex Analysis and Integral Transforms (MAT00003I)

## Module will run

Occurrence Teaching cycle
A Spring Term 2018-19

## Module aims

• To show how analytic properties of the Riemann zeta function lead to results on prime numbers.

• To understand properties of Dirichlet L-functions, and their relation to primes in arithmetic progressions.

## Module learning outcomes

At the end of the module you should be able to:

• To understand how complex analytic techniques enable number theoretic results to be deduced.

• To know some of the analytic properties of the Riemann zeta function

• Appreciate the connection between zeros of the Riemann zeta function and properties of prime numbers.

• To know the prime number theorem and the relation between its error term to the location of zeros of the zeta function.

• To understand Dirichlet characters and analytic properties of Dirichlet L-functions

• To know Dirichlet’s theorem on primes in arithmetic progressions

## Module content

[Pre-requisite knowledge for MSc students: good knowledge of calculus; first courses in both real and complex analysis.]

Syllabus

• Arithmetic functions, partial summation, Dirichlet series and Euler products.
• The Riemann zeta function, its functional equation and analytic continuation. Evaluation at even integers.
• Zeros of the zeta function, and the Riemann Hypothesis.
• The Prime Number Theorem.
• Dirichlet characters and Gauss sums.
• Dirichlet L-functions, their analytic continuation and functional equations.
• Primes in arithmetic progressions.

## Assessment

Task Length % of module mark
University - closed examination
Analytic Number Theory
2 hours 100

None

### Reassessment

Task Length % of module mark
University - closed examination
Analytic Number Theory
2 hours 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.