arXiv subjects

Pierre-Alexandre Bazin

Publications and source records attributed to Pierre-Alexandre Bazin.

2 recordsLinked to original sources

arXiv2025

Bounded Exponential Sums with Multiplicative Coefficients

We investigate when the exponential sum $S_f(x,\alpha) := \sum_{n\le x}f(n)\mathrm{e}(n\alpha)$ is bounded, for a multiplicative function $f$ and $\alpha\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,\alpha)$ is bounded only when $f$ is very close to a twisted Dirichlet character $\chi(n)n^{it}$. We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when $f$ is completely multiplicative and $\alpha$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $\alpha$.

math.NT↗

arXiv2024

Irreducibility of random polynomials of $\mathbb{Z}[X]$

In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $\mu_j,$ then $A$ is irreducible with probability tending to $1$ as $n\to\infty$ under a condition of near-uniformity of the $\mu_j$ modulo four primes (which notably happens when the $\mu_j$ are uniform over a segment of $\mathbb Z$ of length $N\ge 35.$) We improve here the speed of convergence when we have the near-uniformity condition modulo more primes. Notably, we obtain the optimal bound $\mathbb{P}(A \text{ irreducible}) = 1 -O(1/\sqrt n)$ when we have near-uniformity modulo twelve primes, which notably happens when the $\mu_j$ are uniform over a segment of length $N\ge 10^8.$

math.NT↗