arXiv subjects
Publications and source records attributed to Stéphane R. Louboutin.
In 2024, M. K. Ram proved that the class number of an imaginary cyclic quartic number field is never equal to a prime $p\equiv 3\pmod 4$. Here we greatly generalize this result to the case of the non-quadratic imaginary cyclic number fields of $2$-power degrees and not necessarily prime class numbers.
An explicit formula for the mean value of $\vert L(1,χ)\vert^2$ is known, where $χ$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(ζ_p)$ follow. Lately the authors obtained that the mean value of $\vert L(1,χ)\vert^2$ is asymptotic to $π^2/6$, where $χ$ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\pmod{2d}$ which are trivial on a subgroup $H$ of odd order $d$ of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll\frac{\log p}{\log\log p}$. Bounds on the relative class number of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(ζ_p)$ follow. Here, for a given integer $d_0>1$ we consider the same questions for the non-primitive odd Dirichlet characters $χ'$ modulo $d_0p$ induced by the odd primitive characters $χ$ modulo $p$. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,χ')\vert^2$ is asymptotic to $\frac{π^2}{6}\prod_{q\mid d_0}\left (1-\frac{1}{q^2}\right )$, where $χ$ runs over all odd primitive Dirichlet characters of prime conductors $p$ which are trivial on a subgroup $H$ of odd order $d\ll\frac{\log p}{\log\log p}$. As a consequence we improve the previous bounds on the relative class number of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(ζ_p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on $d$ is essentially sharp.