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

Math 401 Fall 2018                 Math 421 Fall 2004                 Math 465 Spring 2013                 Math 467 Fall 2017 

Math 567 Fall 2008                 Math 568 Spring 2018             Math 571 Fall 2018                     Math 597e Spring 2008                  

Math 504 Spring 2009            Math 572 Spring 2010 (old syllabus)                                           Math 597b Spring 2015              

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