Number theory 2
|2 + 0||2 + 0||exam||pure math.||mm1c1se4m||4||optional|
Syllabus designed by:
We illustrate the structure of modern number theory.
- G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. Online edition.
- Elements of analytic number theory and prime number theory, Definition of the Riemann zeta function for s>1, its Euler product representation, the existance of infiniely many primes. Large gaps between primes. Dirichlet theorem on primes in arithmetic progressions (without proof), special cases.
- Diophantine equations. The equation x2+y2=n, Gaussian integers. Other quadratic extensions, there exists a quadratic extension without unique prime factorization. Representation as the sum of 3, resp 4 squares, Lagrange's theorem. Waring's problem, g(k), G(k), lower bounds for them. Pell equations.
- Diophantine approximation, Dirichlet's approximation theorem.
- Elements of algebraic number theory. Algebraic and transcendental numbers, Liouville's theorem, the construction of a transcendental number.
- Elements of combinatorial number theory, Sidon sets. The method of generating functions, the Fibonacci numbers. The elements of the geometry of numbers, the circle problem. The estimate of ∑ 1/p (p prime).