Pre-requisite knowledge for MSc students: good knowledge of calculus; first courses in both real and complex analysis.
Module will run
Occurrence
Teaching period
A
Autumn Term 2021-22
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
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
Online Exam -less than 24hrs (Centrally scheduled) Analytic Number Theory
2 hours
100
Special assessment rules
None
Reassessment
Task
Length
% of module mark
Online Exam -less than 24hrs (Centrally scheduled) 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.
Indicative reading
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)
Mezzadri and Snaith (eds), Recent Perspectives in Random Matrix Theory and Number Theory, LMS Lecture Notes Series, 322. (S 2.81 MEZ)