Georgia Institute of Technology College of Sciences

Large sieve inequalities are a fundamental tool used to investigate prime numbers and exponential sums. I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias first observed by Kummer in 1846. One important byproduct of my work is a proof that

Heath-Brown's famous cubic large sieve is sharp, contrary to popular belief.  This sheds light on some of the mysteries surrounding large sieve inequalities for certain families of arithmetic harmonics and gives strong clues on where to look next for further progress. This is based on joint work with Maksym Radziwill.