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Introduction

This course is a continuation of Math 567 Number Theory I given in the Fall.  However, it is sufficiently self contained than any courses of elementry number theory and abstract algebra would be sufficient prerequisite.  
    The primes are the basic building blocks for the multiplicative structure of the integers and their frequency and density is fundamental.  One of the underlying motives for the course is an amazing and beautiful explicit formula, discovered by

connecting the prime numbers with the zeros of the Riemann zeta-function.  The Riemann Hypothesis concerning the zeros is now the most important unsolved problem in mathematics.  For an account of this and connected questions see the article by Enrico Bombieri at http://www.claymath.org/prizeproblems/riemann.htm.   We will follow the  very first tentative steps taken in this direction by Chebyshev, Mertens, Hadamard and de la Vallée Poussin.

                                                

Pafnuty Lvovich Chebyshev  Franz Carl Joseph Mertens  Jacques Salomon Hadamard
1821-1894                           1840 - 1927                         1865 - 1963

 

               Charles Jean Gustave NicolasBaron de la Vallée Poussin, 1866 - 1962

Closely interwoven with the Riemann zeta function and its generalisations are theta functions and modulur functions, first investigated in detail by Jacobi.  

Carl Gustav Jacob Jacobi, 1804 - 1851

    Other fundamental questions often arise from the study of the solubility of diphantine equations.  Sometimes the solution of a diophantine equation can be made to depend on approximations to real numbers by rationals.  An example is Pells' equation x²-dy²=1.  One simple theorem on such approximations is due to Dirichlet and involves the first use of the box principle. This also leads to Liouville's theorem

                                                       

Johann Peter Gustav Lejeune Dirichlet, 1805 - 1859       Joseph Liouville, 1809 - 1882

on the approximation of algebraic numbers and enables us to construct simple examples of transcendental numbers.

    Fundamental to number theory is the idea that various generalisations of the ring of integers and the field of rational numbers yield ways of furthering our understanding of the eponymous prototypes.  Thus rings of algebraic integers and algebraic number fieldsneed to be studied as well as Hensel's p-adic number fields.  An interesting principle connected with the latter is Hasse's

                                

Kurt Hensel, 1861 - 1941            Helmut Hasse, 1898 - 1979

observation that solubility in all completions of the rationals is sometimes sufficient to ensure solubility in the rationals.

• Chebychev's inequalities for the prime counting function. and Merten's theorem. 
• Riemann's zeta functions and the prime number theorem of Hadamard and de la Vallée Poussin. 
• Modular functions. 
• The approximation of real numbers by rationals, criteria for irrationality, examples of transcendental numbers. 
• Algebraic Numbers and Algebraic Integers. 
• Ideal class groups. 
• p-adic numbers of Hensel and the Hasse principle. 
• Finite Fields. 
• Jacobi sums. 
• Equations over finite fields. 
• An Introduction to Elliptic curves

• DETAILS

• Professor: Robert C. Vaughan
• Office: 335 McAllister
• Telephone: 865-3583
• Email: rvaughan@math.psu.edu
• Office Hours: MTW 1:45-2:15 and otherwise by arrangement.
• Class: TR 2:30-3:45, 335 McAllister.
• Schedule Number: 770467