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Analytic Number Theory

Unit code: MATH41022
Credit Rating: 15
Unit level: Level 4
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N



Additional Requirements

MATH41022 pre-requisites

Students must have taken MATH20101 OR MATH20142

Students are not permitted to take MATH41022 and MATH61022 for credit in an undergraduate programme and then a postgraduate programme.


To show how the tools of Mathematical analysis can be used to prove results about prime numbers and functions defined on the integers.


We start by giving two proofs of the infinitude of primes. The methods are elementary but poor in that they do not tell us the truth of how many primes there are. Stronger tools are introduced, improving the results until we can give a proof of the Prime Number Theorem.

Learning outcomes

On completion of this unit students will be able to: 

  • prove elementary results on sums over primes and use these to calculate averages of additive  arithmetic  functions,
  • utilise the correspondence between the product of Dirichlet series and convolution of arithmetic functions to factor multiplicative functions and then calculate their averages,
  • prove some analytic properties of the Riemann zeta function, including an analytic continuation, a zero-free region, and estimates on the growth of the zeta function,
  • prove the Prime Number Theorem with an error term.


Assessment Further Information

End of semester examination: three hours weighting 100% (MATH41002)


  • Two proofs of the infinitude of primes. [3]
  • Arithmetic functions and Dirichlet Series. [5]
  • Elementary Prime Number Theory. [6]
  • The Prime Number Theorem. [8]
  • Sums of arithmetic functions. [6]

Recommended reading

  • T. Apostol, Introduction to Analytic Number Theory, 1st edition. 1976, Corrected 5th edition 2010, Springer, 1441928057
  • G.J.O. Jameson, The Prime Number Theorem, LMS Student Texts 53, CUP 2003, 0-521-89110-8.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 33 hours
  • Tutorials - 11 hours
  • Independent study hours - 106 hours

Teaching staff

Mark Coleman - Unit coordinator

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