Geometric & Analytic Number Theory - MAT00109M

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Module summary

This module comes in two parts. The first part provides a deeper and more quantitative understanding of the structure of the real numbers through Diophantine approximation, and helps illustrate the interplay of different branches of mathematics by the use of algebra, probability and basic results from the theory of Lebesgue measure. The second part continues in that theme, and shows how complex-analytic properties of the Riemann zeta function lead to deep results on prime numbers.

Related modules

Additional information

For MSc students: A course on elementary number theory and a course on complex analysis.

Module will run

Occurrence	Teaching period
A	Semester 1 2023-24

Module aims

This module comes in two parts. The first part provides a deeper and more quantitative understanding of the structure of the real numbers through Diophantine approximation, and helps illustrate the interplay of different branches of mathematics by the use of algebra, probability and basic results from the theory of Lebesgue measure. The second part continues in that theme, and shows how complex-analytic properties of the Riemann zeta function lead to deep results on prime numbers.

Module learning outcomes

At the end of this modules, students will be able to:

1. Use a range of ideas and techniques in Diophantine approximation.

2. Use algebraic and probabilistic ideas in the context of metric number theory.

3. Apply complex analytic techniques to deduce number theoretic results.

4. Derive some of the analytic properties of the Riemann zeta function.

5. Utilise the relationship between zeros of the Riemann zeta function and properties of prime numbers, specifically in the context of the proof of the Prime Number Theorem.

Module content

• Dirichlet’s approximation theorem, and Diophantine approximation in higher dimensions.

• Khintchine’s theorem and the zero-one law.

• Analytic properties of Dirichlet series and Euler products, especially the Riemann zeta function.

• Zeros of the Riemann zeta function.

• Proof of the Prime Number Theorem.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	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.

Indicative reading

• G H Hardy and E M Wright, The Theory of Numbers, Oxford University Press.

• Apostol, Introduction to Analytic Number Theory, (Springer-Verlag New York)

• Miller and Takloo-Bighash, An Invitation to Modern Number Theory, (Princeton University Press) (S 2.81 MIL)

• Edwards, Riemann's Zeta Function (Dover Publications) (S 7.36 EDW)

• Iwaniec and Kowalski, Analytic number theory (American Mathematical Society) (S 2.81 IWA)

• Montgomery and Vaughan, Multiplicative number theory I : classical theory (Cambridge University Press) (S 2.814 MON)