Fonctions complètement $Q$-additives le long des polynômes irréductibles à coefficients dans un corps fini
Volume 203 / 2022
Acta Arithmetica 203 (2022), 97-136
MSC: 11T06, 11T23, 11T55.
DOI: 10.4064/aa191128-5-2
Published online: 27 April 2022
Abstract
Let $\mathbb F$ be a finite field with $q$ elements, ${\bf I}_{N}$ the set of monic irreducible polynomials of degree $N$ over $\mathbb F$, $Q \in \mathbb F[T]$ and $w$ an integer-valued completely $Q$-additive function. The goal of this work is to study the exponential sums $\sum _{P\in {\bf I}_{N}}\mathrm {exp}(2\pi i\alpha w(P))$ for $\alpha \in \mathbb R $. In particular, we deduce from this study a sufficient condition on $w$ under which, for any $(a, m) \in \mathbb N \times ({\mathbb N }\setminus \{0,1\})$, we have $$ {\rm Card}\, \{P\in {{\bf I}_{N} } \,; w(P)\equiv a\ ({\rm mod}\, m)\} = \frac {q^{N}}{mN} + O( q^{(1-h)N} )$$ with $0 \lt h \lt 1$.
