Course Notes

Advanced probabilistic methods applied to arithmetic functions, sieve theory, and the study of distribution of prime numbers using analytic techniques.

Rigorous introduction to enumerative combinatorics, graph theory, and partially ordered sets. Focus on structural understanding and bijective proofs.

The symbolic method for generating functions. Isomorphisms between combinatorial classes and analytic properties of their generating series.

Graduate-level treatment of holomorphic functions, Cauchy's theorem, singularities, and conformal mappings.

Analytic continuation, functional equation, distribution of prime numbers, and the connection between zeros and primes.

Advanced topics including elliptic functions, modular forms, infinite products, and the Riemann mapping theorem.

Number fields, rings of integers, ideals, unique factorization of ideals, and the ideal class group.

Residue theory, evaluation of improper integrals, argument principle, and harmonic functions.

Topology of metric spaces, sequences and series, continuity, and differentiability in rigorous settings.

Complex numbers, holomorphic functions, Cauchy-Riemann equations, and power series expansions.

Multivariable calculus: partial derivatives, gradient, multiple integrals, and vector fields.

Foundations of single-variable calculus: limits, continuity, the derivative, and introduction to integration.