Instructor: Dr . Szilárd Gy. Révész

Text: Selected chapters of Melvyn B. Nathanson: Elementary Methods in Number Theory, Graduate Texts in Mathematics
vol. 195, Springer Verlag, 2000. In case of departing from the book, notes will be handed out.

Prerequisite: General mathematical experience of the undergraduate level is expected. Notions, methods and techniques from elementary algebra (Abelian groups, vector spaces, systems of linear equations etc.), analysis (limit, integration, infinite series etc.), elements of probability theory (finite probability space, combinatorial computation of probabilities, expectation, variance), and first course of number theory are used throughout. However, the basic concepts of the applied theorems will always be explained. A course in complex function theory (analytic functions, Taylor series, Laurent series, complex line integrals, Cauchy's integral theorems, Theorem of Residues), equivalent of CLX, prior or parallel to this course is essential.
 

Course description: The course provides an introduction to some topics of number theory. The aim is to highlight the inner beauty of the topic and the richness of general methods and deep problems in number theory. Particular emphasis is put on the interplay between number theory and other branches of mathematics, present both in the variety of mathematical techniques and theories used, and in the various developments generated by number theory.

Topics:

Short review. Basic notions and theorems about divisibility, primes and congruences. Definitions of the most frequently used number theoretic functions.

Congruences. System of simultaneous congruences. Order, primitive root. Quadratic congruences, quadratic residues, Legendre symbol. The ``little'' Fermat theorem. The quadratic reciprocity law.

Arithmetical functions. Euler`s f(n), the divisor function d(n), d<sub>k</sub>(n). s(n) and perfect numbers. The number of prime divisors w(n),  W(n). Möbius`s function m(n), Liouville's l(n), von Mangoldt's L(n). Multiplicative functions and additive functions. Operations and identities in the set of arithmetical functions. Convolution, Möbius inversion. Partial summation, elementary manipulations. Estimates on the number of primes. Average of some arithmetic functions.

Additive and multiplicative functions.  Probability of (m,n)=1. Probabilistic heuristics and the Turán-Kubilius inequality. A theorem of Erdõs: characterization of the logarithm. Mean values of multiplicative functions.

Analytic number theory. Dirichlet series. Analytic formulations of number theoretic identities. The Riemann zeta function z(s). A characterization of z(s). Analytic properties of z (s). A most famous problem of the XXI^st century: the Riemann Conjecture. The Riemann--von Mangoldt Formula. The Riemann Hypothesis and the Prime Number Theorem. The Divisor Problem. Dirichlet's Theorem about primes in arithmetic progressions.

If time permits, we will have an outlook of additive number theory, Diophantine approximation, and algebraic and transcendence numbers.

An outlook of additive number theory. The Goldbach problem. Vinogradov's three primes theorem. Waring's problem: g(k) and G(k), lower bounds.

Diophantine approximation  and  algebraic numbers. Approximation of  irrationals  by rationals. Minkowski's Theorem. Properties of algebraic numbers and   the  minimal polynomial. Liouville's theorem  on  the  approximation of  algebraic numbers and construction of transcendental  numbers. Algebraic integers.

Transcendece theory. Algebraic and transcendental numbers, e and  p iare transcendental.