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

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) 

Some Photographs

Some Quotations                   Pronunciation of British Names

Obituary of Thomas Vaughan
    Tribute by Robert Reeves

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 Fall 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                  A theorem of E. M. Wright on Waring's Problem   Integer points on elliptic curves-II            
Uniform distribution