arXiv subjects

Stéphane Louboutin

Publications and source records attributed to Stéphane Louboutin.

6 recordsLinked to original sources

arXiv2025

Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials

We fill the gaps in A. Gica's determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\leq\sqrt{d}$. We also determine all the even positive integers $d$ for which the number of distinct prime divisors of $f_d(x)$ is less than or equal to $2$ for all the positive and even integers $x\leq\sqrt{d}$. These problems are related to the famous Frobenius-Rabinowitsch's characterization of the imaginary quadratic number fields ${\mathbb Q}(\sqrt{-d})$ of odd discriminants with class number one in terms of the primality of $f_d(x)/4$ for all the positive and odd integers $x\leq\sqrt{d}$. However, the solution to our problem is much more difficult to come up with. We also begin to address the same problems for the case of $f_d(x)=d-x^2$, in relation with the class groups of the real quadratic number fields ${\mathbb Q}(\sqrt{d})$.

math.NT↗

arXiv2025

An asymptotic on the logarithms of the relative class numbers of imaginary abelian number fields of prime conductor and large degree

An asymptotic on the logarithms of the relative class numbers of the cyclotomic number fields of prime conductors $p$ is known. Here we give an asymptotic on the logarithms of the relative class numbers of the imaginary abelian number fields of prime conductors $p$ and large degrees $m =(p-1)/d$ with $ϕ(d)=o(\log p)$. We also show that this asymptotic does not hold true anymore under the only slightly weaker restriction $ϕ(d)=O(\log p)$.

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arXiv2024

Dedekind sums and mean square value of $L(1,χ)$ over subgroups

An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo not necessarily prime moduli $f>2$ that are trivial on a subgroup $H$ of the multiplicative group $({\mathbb Z}/f{\mathbb Z})^*$. This explicit formula involves summation $S(H,f)$ of Dedekind sums $s(h,f)$ over the $h\in H$. A result on some cancelation of the denominators of the $s(h,f)$'s when computing $S(H,f)$ is known. Here, we prove that for some explicit families of $f$'s and $H$'s this known result on cancelation of denominators is the best result one can expect. Finally, we surprisingly prove that for $p$ a prime, $m\geq 2$ and $1\leq n\leq m/2$, the values of the Dedekind sums $s(h,p^m)$ do not depend on $h$ as $h$ runs over the elements of order $p^n$ of the multiplicative cyclic group $({\mathbb Z}/p^m{\mathbb Z})^*$.

math.NT↗

arXiv2024

Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers

Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{ϕ(f)}\sum_{χ(-1)=(-1)^m}\vert L(m,χ)\vert^2,$$ where $χ$ ranges over the $ϕ(f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,χ_1)\cdots L(m_{n-1},χ_{n-1})\overline{L(m_n,χ_1\cdotsχ_{n-1})}$.

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arXiv2023

Dedekind sums and class numbers of imaginary abelian number fields

As a consequence of their work, Bruce C. Berndt and Ronald J. Evans in 1977 and Larry Joel Goldstein and Michael Razar in 1976 obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. Here we focus only on this formula and give a new short and simple proof of it. We finally express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums.

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arXiv2020

Second moment of Dirichlet $L$-functions, character sums over subgroups and upper bounds on relative class numbers

We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E. Elma. We obtain an asymptotic formula for ${\cal A}(p,d)$ which holds true for any divisor $d$ of $p-1$ removing previous restrictions on the size of $d$. This anwers a question raised in Elma's paper. Our proof relies both on estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application we deduce the following bound $h_{p,d}^- \leq 2\left (\frac{(1+o(1))p}{24}\right )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod d$ and degree $m=(p-1)/d$.

math.NT↗