arXiv subjects

Théo Untrau

Publications and source records attributed to Théo Untrau.

4 recordsLinked to original sources

arXiv2026

A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races

We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races.

math.NT↗

arXiv2025

Wasserstein metrics and quantitative equidistribution of exponential sums over finite fields

The Wasserstein distance between probability measures on compact spaces provides a natural invariant quantitative measure of equidistribution, which is partly similar to the classical discrepancy appearing in Erdös-Turán type inequalities in the case of tori, but is a more intrinsic quantity. We recall the basic properties of Wasserstein distances and present applications to quantitative forms of equidistribution of exponential sums in two examples, one related to our previous work on the equidistribution of ultra-short exponential sums, and the second a quantitative form of the equidistribution theorems of Deligne and Katz.

math.NT↗

arXiv2023

Ultra-short sums of trace functions

We generalize results of Duke, Garcia, Hyde, Lutz and others on the distribution of sums of roots of unity related to Gaussian periods to obtain equidistribution of similar sums over zeros of arbitrary integral polynomials. We also interpret these results in terms of trace functions, and generalize them to higher rank trace functions.

math.NT↗

arXiv2021

Equidistribution of exponential sums indexed by a subgroup of fixed cardinality

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We study distribution properties of these families of sums as $q$ grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of such regions.

math.NT↗