Course Materials
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- Lecturer: Profile: Ben Green
The only essential prerequisite is ASO Number theory. Some basic familiarity with complex analysis is useful in some places, thus A2.2 and A.4 are desirable (but not essential) prerequisites. Basic probability theory as covered in M.3 will be needed at a couple of points.
The aim of the course is to present a selection of classic topics in number theory. Whilst the highlights are interesting and elegant in their own right, they also provide relevant background for further study. A further aim is to showcase the variety of techniques which have been used to study properties of the integers, from analysis to probability theory
To gain an appreciation for, and an understanding of, a variety of classic topics in number theory. To realise that number theory draws upon techniques from many other parts of pure mathematics.
Irrationality, approximation and continued fractions. Continued fractions of rationals and Euclid's algorithm. Dirichlet's theorem. Best approximations. Continued fractions of quadratic irrationals. Pell's equation. Irrationality of e. Construction of a transcendental number. [4 lectures]
Arithmetic functions and their distribution. The functions τ (number of divisors), ω (number of prime factors) and ϕ (Euler's totient function). Mean value of τ and the hyperbola method. Hardy-Ramanujan law and normal order of ω and τ . Divisor bound, mean value of ϕ and the probability that random integers are coprime. Introduction to Dirichlet series. [3 lectures]
Distribution of primes. Upper and lower bounds on π(X) (Chebyshev bounds). The Riemann ζ-function and Euler's proof of infinitely many primes. Analytic continuation to ℜs > 0 and statement of the Riemann Hypothesis. Discussion of the prime number theorem and consequences of the Riemann Hypothesis. Dirichlet characters and L-functions. Dirichlet's proof that there are infinitely many primes a mod q when a, q are coprime, assuming L(1, χ) ≠ 0. Proof that L(1, χ) ≠ 0 in special cases. [5 lectures]
Binary quadratic forms and sums of squares. Basic reduction theory of positive definite binary forms, class number. Review of quadratic reciprocity. Representation of primes by positive definite binary quadratic forms. Brief discussion of class number formula. Lagrange's theorem on sums of four squares. [4 lectures]