This note describes the
following topics: Pythagorean Triples, Quadratic Rings, Quadratic Reciprocity,
The Mordell Equation, The Pell Equation, Arithmetic Functions, Asymptotics of
Arithmetic Functions, The Primes: Infinitude, Density and Substance, The Prime
Number Theorem and the Riemann Hypothesis, The Gauss Circle Problem and the
Lattice Point Enumerator, Minkowski’s Convex Body Theorem, The Chevalley-Warning
Theorem, Dirichlet’s Theorem on Primes in Arithmetic Progressions, Rational
Quadratic Forms and the Local-Global Principle.
This lecture note is
an elementary introduction to number theory with no algebraic prerequisites.
Topics covered include primes, congruences, quadratic reciprocity, diophantine
equations, irrational numbers, continued fractions, and partitions.
This note covers the following topics: Divisibility and
Primes, Congruences, Congruences with a Prime-Power Modulus, Euler's Function
and RSA Cryptosystem, Units Modulo an Integer, Quadratic Residues and Quadratic
Forms, Sum of Powers, Fractions and Pell's Equation, Arithmetic Functions, The
Riemann Zeta Function and Dirichlet L-Function.
The
notes contain a useful introduction to important topics that need to be
addressed in a course in number theory. Proofs of basic theorems are presented
in an interesting and comprehensive way that can be read and understood even by
non-majors with the exception in the last three chapters where a background in
analysis, measure theory and abstract algebra is required.
This note contains the
following subtopics: Classfield theory, homological formulation, harmonic
polynomial multiples of Gaussians, Fourier transform, Fourier inversion on
archimedean and p-adic completions, commutative algebra: integral extensions
and algebraic integers, factorization of some Dedekind zeta functions into
Dirichlet L-functions, meromorphic continuation and functional equation of zeta,
Poisson summation and functional equation of theta, integral representation of
zeta in terms of theta.
Robert Daniel Carmichael (March
1, 1879 – May 2, 1967) was a leading American mathematician.The purpose of this
little book is to give the reader a convenient introduction to the theory of
numbers, one of the most extensive and most elegant disciplines in the whole
body of mathematics. The arrangement of the material is as follows: The five
chapters are devoted to the development of those elements which are essential to
any study of the subject. The sixth and last chapter is intended to give the
reader some indication of the direction of further study with a brief account of
the nature of the material in each of the topics suggested.