Exploration of the exponential sum over primes and over sums of two squares

Time: 10:30 - 11:30, June 24, 2026

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: The trigonometric polynomial on the primes $T(\alpha)=\sum_{p\le N} e(p\alpha)$ contains many information on the primes, though it is still very badly understood. During this lecture we shall present several pictures and almost-videos to describe the behaviour of this quantity.
We shall successively explore:
(1) The places where it takes large values,
(2) Pointwise bounds,
and (3) Pointwise bounds for the cousin $U$ of $T$ defined by $U(\alpha)=\sum_{b\le N}e(b\alpha)$ where $b$ is an integer that can be written as a sum of two squares. This talk should be accessible for a number theory audience, though some small parts may remain obscure to the non-specialist.