Analytic Number Theory Lecture Notes by Andreas Strombergsson

Analytic Number Theory Lecture Notes by Andreas Strombergsson

Analytic Number Theory Lecture Notes by Andreas Strombergsson

This note covers the following topics: Primes in
Arithmetic Progressions, Infinite products, Partial summation and Dirichlet
series, Dirichlet characters, L(1, x) and class numbers, The distribution of the
primes, The prime number theorem, The functional equation, The prime number
theorem for Arithmetic Progressions, Siegel’s Theorem, The Polya-Vinogradov
Inequality, Sums of three primes, The Large Sieve, Bombieri’s Theorem.

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.

Analytic
number theory provides some powerful tools to study prime numbers, and most of
our current knowledge of primes has been obtained using these tools. Topics
covered includes: Primes and the Fundamental Theorem of Arithmetic, Arithmetic
functions: Elementary theory, Dirichlet series and Euler products and Asymptotic
estimates, Distribution of primes: Elementary results and Proof of the Prime
Number Theorem, Primes in arithmetic progressions.

This is a
textbook about classical elementary number theory and elliptic curves. The first
part discusses elementary topics such as primes, factorization, continued
fractions, and quadratic forms, in the context of cryptography, computation, and
deep open research problems. The second part is about elliptic curves, their
applications to algorithmic problems, and their connections with problems in
number theory.

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.

This note covers the following topics: Algebraic numbers and algebraic
integers, Ideals, Ramification theory, Ideal class group and units, p-adic
numbers, Valuations, p-adic fields.

This
note explains the following topics:
Algebraic numbers, Finite continued fractions, Infinite continued fractions,
Periodic continued fractions, Lagrange and Pell, Euler’s totient function,
Quadratic residues and non-residues, Sums of squares and Quadratic forms.