MAS430 Analytic Number Theory

Semester 1, 2021/22 10 Credits
Lecturer: Dr Tobias Berger Timetable Reading List
Aims Outcomes Teaching Methods Assessment Full Syllabus

Analytic number theory aims to study number theory by using analytic tools (inequalities, limits, calculus, etc). In this course we will mainly focus on studying the distribution of prime numbers by using analysis. Perhaps it is surprising that such a link even exists!

We will try and give precise answers to questions such as; Roughly how big is the nth prime? Approximately how many primes are less than a given number? How fast does the sequence of primes diverge to infinity? Is there always a prime between n and 2n?
A big result to be proved in the course is Dirichlet's theorem on primes in arithmetic progressions. This tells us that for coprime a,b the sequence an+b contains infinitely many primes. The general proof of this result features a unique blend of algebra and analysis and was the cornerstone of 19th century number theory.

Prerequisites: MAS114 (Numbers and Groups); MAS332 (Complex Analysis)
No other modules have this module as a prerequisite.

Outline syllabus

  • Distribution of primes
  • Implications of the Prime Number Theorem
  • Dirichlet series
  • The Riemann zeta function
  • Dirichlet's Theorem on primes in arithmetic progression


Aims

  • To illustrate how general methods of analysis can be used to obtain results about integers and prime numbers
  • To investigate the distribution of prime numbers
  • To consolidate earlier knowledge of analysis through applications

Learning outcomes

  • Understand better the distribution of prime numbers
  • Know the basic theory of zeta- and L-functions
  • Understand the proof of Dirichlet's Theorem

Teaching methods

Lectures, problem solving


20 lectures, no tutorials

Assessment

One formal 2.5 hour written examination. Format: 4 questions, answer all.

Full syllabus

1. Distribution of prime numbers
(6 lectures)

Multiple proofs of the infinitude of primes, for example using the divergence of ∑[1/p]. Classical proofs of infinitude of primes in arithmetic progressions of small common difference. Statements of Dirichlet's theorem and the Green-Tao theorem. Bertrand's Postulate with proof. The prime counting function π(x) and certain upper/lower bounds, for example Chebyshev's inequalities. Statement of Prime Number Theorem with applications.
2. Arithmetic functions and Dirichlet series
(5 lectures)
Arithmetic functions, Dirichlet convolution and Mobius Inversion. Multiplicative and completely multiplicative functions. The Dirichlet series attached to an arithmetic function. Euler products for Dirichlet series of multiplicative and completely multiplicative functions. Convergence and absolute convergence of Dirichlet series in a half-plane.
3. The Riemann zeta function
(5 lectures)
The zeta function and convergence for Re(s) > 1. Other properties including behaviour on the real line. Bernoulli numbers and Bernoulli polynomials. Evaluation of ζ(2k) using Bernoulli polynomials and Fourier series. Remarks on ζ(2k+1). Analytic continuation of ζ(s) to Re(s) > 0. Remarks on the Riemann Hypothesis.
4. Dirichlet's Theorem
(4 lectures)
Toy example of Dirichlet's proof for primes mod 4. Dirichlet characters and orthogonality. Dirichlet L-functions and their convergence. Proof of Dirichlet's Theorem.

Reading list

Type Author(s) Title Library Blackwells Amazon
B Apostol Introduction to analytic number theory 512.81 (A) Blackwells Amazon
B Chan Heng Huat Analytic Number Theory for Undergraduates Blackwells Amazon
B Davenport Multiplicative number theory 512.81 (D) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.