Colloquium

Igor Shparlinski, University of New South Wales

"Integers of prescribed arithmetic structure in residue classes"

We give an overview of recent results about the distribution of some special integers in residues classes modulo a large integer $q$. Questions of this type were introduced by Erdos, Odlyzko and Sarkozy (1987), who considered products of two primes as a relaxation of the classical question about the distribution of primes in residue classes. Since that time, numerous variations have appeared for different sequences of integers. The types of numbers we discuss include smooth, square-free, square-full and almost primes integers.

We also expose, without going into technical details, the wealth of different techniques behind these results: sieve methods, bounds of short Kloosterman sums, bounds of short character sums and many others.

MC 5501