A model problem for multiplicative chaos in number theory
arXiv:2108.07264 · doi:10.4171/LEM/1031
Abstract
Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson's conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area.
arXiv comment25 pages
References in corpus (7)
- Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function
- On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
- Moments of the Riemann zeta function on short intervals of the critical line
- The Fyodorov-Hiary-Keating Conjecture. I
- Permutations fixing a k-set
- Secular Coefficients and the Holomorphic Multiplicative Chaos
- Maxima of log-correlated fields: some recent developments