published on May 19, 2026
Sonification in number theory: listening to the Riemann zeta function
By Jonathan Love
A dichotomy in the tail behaviour of quadratic Weyl sums
Appears in collection : 2026 - T1 - WS3 - Integrating Research and Illustration in Number Theory
Jointly with Tariq Osman, we completed the classification of the tail behaviour of the limiting distributions of all quadratic Weyl sums of the form $1/\sqrt{N} \sum_{n=1}^N e( ((1/2)n^2+\beta n)x+\alpha n)$. When $\alpha$ and $\beta$ are both rational, while trying to understand the contribution of certain orbits to the heavy tails, we discovered that some pairs actually lead to a compactly supported limiting distribution. I will especially emphasise the role of mathematical illustration in our understanding of the geometry of the relevant orbits, as well the importance of numerical simulations to validate our results and prompt new questions.
