Stéphane R. Louboutin

THE BRAUER–SIEGEL THEOREM

Explicit bounds are given for the residues at of the Dedekind zeta functions of number fields. As a consequence, a simple proof of the Brauer-Siegel theorem and explicit lower bounds for class numbers of number fields are obtained. Compared with Starks original approach, the paper is explicit and more satisfactory for number fields containing quadratic subfields. Examples are given of fully explicit lower bounds for class numbers of various types of number fields, for example normal and non-normal number fields of odd degree, with an emphasis on cubic fields, real cyclic quartic number fields, and non-normal quartic number fields containing an imaginary quadratic subfield.

A twisted quadratic moment for Dirichlet L-functions

Given c, a positive integer, we give an explicit formula for the quadratic moments where X-f is the set of the odd Dirichlet characters mod f with f > 2.

Mean values of L-functions and relative class numbers of cyclotomic fields

Using formulas for quadratic mean values of L-functions at s = 1, we recover previously known explicit upper bounds on relative class numbers of cyclotomic elds. We also obtain new better bounds.

On the divisibility of the class number of imaginary quadratic number fields

We prove that if at least one of the prime divisors of an odd integer U > 3 is equal to 3 mod 4, then the ideal class group of the imaginary quadratic field Q( √1 - 4U n ) contains an element of order n.

On the fundamental units of a totally real cubic order generated by a unit

We give a new and short proof of J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler following a recent result: if is a totally real cubic algebraic unit, then there exists a unit η ∈ Z[ ] such that { , η} is a system of fundamental units of the group U of the units of the cubic order Z[ ], except for an infinite family for which is a square in Z[ ] and one sporadic exception. Not only is our proof shorter, but it enables us to prove a new result: if the conjugates ′ and ′′ of are in Z[ ], then the subgroup generated by and ′ is of bounded index in U , and if > 1 > | ′| ≥ | ′′| > 0 and if ′ and ′′ are of opposite sign, then { , ′} is a system of fundamental units of U .

Explicit upper bounds for values at s=1 of Dirichlet L-series associated with primitive even characters

Abstract Let S be a given finite set of pairwise distinct rational primes. We give an explicit constant κ S such that for any even primitive Dirichlet character χ of conductor q χ >1 we have ∏ p∈S 1− χ(p) p L(1,χ) ⩽ 1 2 ∏ p∈S 1− 1 p ( log q χ +κ S )+o(1), where o (1) is an explicit error term which tends rapidly to zero when q χ goes to infinity.

The Class Number One Problem for Some Non-Normal Sextic CM-Fields

We determine all the non-normal sextic CM-fields (whose maximal totally real subfields are cyclic cubic fields) which have class number one. There are 19 non-isomorphic such fields.

Real zeros of Dedekind zeta functions

Building on Stechkin and Kadiris ideas we derive an explicit zero-free region of the real axis for Dedekind zeta functions of number fields. We then explain how this new region enables us to improve upon the previously known explicit lower bounds for class numbers of number fields and relative class numbers of CM-fields.

Resultants of Chebyshev Polynomials: A Short Proof

We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan.

Discriminants of 𝔖n-orders

Let α be an algebraic integer of degree n ≥ 2. Let α1,…,αn be the n complex conjugate of α. Assume that the Galois group Gal(ℚ(α1,…,αn)/ℚ) is isomorphic to the symmetric group 𝔖n. We give a ℤ-basis and the discriminant of the order ℤ[α1,…,αn]. We end up with an open question showing that this problem seems much harder when we assume that ℚ(α)/ℚ is already Galois or even cyclic.