arXiv subjects

Jérémy Dousselin

Publications and source records attributed to Jérémy Dousselin.

3 recordsLinked to original sources

arXiv2024

Complex moments of class numbers with fundamental unit restrictions

We explore the distribution of class numbers $h(d)$ of indefinite binary quadratic forms, for discriminants $d$ such that the corresponding fundamental unit $\varepsilon_d$ is lower than $d^{1/2+α}$, where $0<α<1/2$. To do so we find an asymptotic formula for $z^{th}$-moments of such $h(d)$'s, over $d\leq x$, uniformly for a complex number $z$ in a range of the form $|z|\leq(\log x)^{1+o(1)}$, $\Re(z)\geq -1$. This is achieved by constructing a probabilistic random model for these values, which we will use to obtain estimates for the distribution function of $h(d)$ over our family. As another application, we give an asymptotic formula for the number of $d$'s such that $h(d)\leq H$ and $\varepsilon_d\leq d^{1/2+α}$ where $H$ is a large real number.

math.NT↗

arXiv2023

Zeros of linear combinations of Dirichlet $L$-functions on the critical line

Let $F$ be a linear combination of $N\geq 1$ Dirichlet $L$-functions attached to even (or odd) primitive characters with the same modulus. Selberg proved that a positive proportion of non-trivial zeros of $F$ lie on the critical line. Our work here is to provide an explicit lower bound for this proportion. In particular, we show that the lower bound $2.16\times 10^{-6}/(N\log N)$ is admissible for large $N$.

math.NT↗

arXiv2023

On abelian cubic fields with large class number

We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an explicit lower bound for extreme values of class numbers of abelian cubic fields.

math.NT↗