Prof. R. C. Vaughan FRS Bob Vaughan  Robert Vaughan

Penn State Mathematics Web Page

Research Interests :

My main research is in number theory, that is, the study of the properties of the whole numbers, especially by the use of analytic techniques. Particular subjects of interest are Waring's problem, the Goldbach problem, the Hardy-Littlewood method, the use of "smooth numbers", i.e. numbers without large prime factors, the distribution of prime numbers, exponential sums over integer sequences such as the sequence of primes, properties of the Riemann zeta-function and Dirichlet L-functions and diophantine approximation.

It can be of no practical use to know that π is irrational, but if we can know, it surely would be intolerable not to know.
E. C. Titchmarsh (1899-1963) 
Publications                   Some Photographs                  Some Quotations                   Pronunciation of British Names

Obituary of Thomas Vaughan
    Tribute by Robert Reeves

Algebra and Number Theory Seminar

A Course of Elementary Number Theory
This is a book based on the elementary number theory courses I have taught over nearly fifty years at Imperial College London and Penn State University.  I don't think publishers should charge huge amounts for what can be produced with very little effort.

Math 401 Spring 2019       Math 421 Fall 2004          Math 465 Spring 2021       Math 467 Fall 2017        Math 567 Fall 2008         Math 568 Spring 2020    
Math 571 Spring 2021       Math 597e Spring 2008    Math 504 Spring 2009       Math 572 Spring 2010 (old syllabus)                     Math 597b Spring 2015              

Lagrange's 4 square theorem  Remarks on the Selberg Sieve                   Jarnik's theorem on integer points on curves     Dirichlet's theorem and Farey fractions  
Modular forms I                     The large sieve                                           Brandon Hanson's notes on Stepanov-Burgess  Continued fractions   
Modular forms II                    The Bombieri-A. I. Vinogradov theorem  Khinchin heuristics                                             The Geometry of Numbers
Density and sum sets              The Goldston, Pintz, Yilidirim theorem    Inhomogeneous approximation                          Basic Transcendence theory
Rouché's Theorem                  Uniform distribution                                  A theorem of E. M. Wright on Waring's Problem