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 y-factorable numbers, i.e. numbers with no prime factors exceeding y, 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 2007-, Spring 2007, Fall 2006, Spring 2006, Fall 2004, Spring 2004, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000

A Course of Elementary Number Theory
An Introduction to Analysis
An Introduction to Factorization and Primality Testing
These are books based on the elementary number theory and introductory analysis 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 2024       Math 421 Fall 2004           Math 465 Spring 2025       Math 467 Fall 2024         Math 504 Spring 2009      
Math 568 Spring 2020       Math 571 Spring 2025      Math 597e Spring 2008      Math 597b Spring 2015  Guth-Maynard Study Group        

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                           A theorem of E. M. Wright on Waring's Problem          
Rouché's Theorem                  Uniform distribution                                 Basic Transcendence theory                               A survey of the Montgomery_Hooley Theorem          
The Basel Problem                  Commentary on an Exercise in "The Hardy-Littlewood Method"