# Multiplicative Functions

A multiplicative function $f: {\Bbb N} \to {\Bbb C}$ is a function satisfying $f(mn)=f(m)f(n)$. Many naturally occuring functions in number theory are multiplicative. Over the last several years, Andrew Granville and I have been studying various features of multiplicative functions. I will discuss some aspects of this work. For example, I will answer the question of how many numbers up to a given number $x$ are quadratic residues (you are free to choose the prime $p$ so as to minimize the answer). As another example, I will discuss character sums and a recent improvement of a classical inequality of Polya and Vinogradov.

This talk is part of the Kuwait Foundation Lectures series.