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Zeta functions for the Riemann zeros

André Voros (2003)

Annales de l’institut Fourier

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structures, plus countably many special values) are explicitly displayed.

Zeta functions of Jordan algebras representations

Dehbia Achab (1995)

Annales de l'institut Fourier

This work is about a generalization of Kœcher’s zeta function. Let V be an Euclidean simple Jordan algebra of dimension n and rank m , E an Euclidean space of dimension N , ϕ a regular self-adjoint representation of V in E , Q the quadratic form associated to ϕ , Ω the symmetric cone associated to V and G ( Ω ) its automorphism group G ( Ω ) = { g G L ( V ) | g ( Ω ) = Ω } . ( H 1 ) Assume that V and E have Q -structures V Q and E Q respectively and ϕ is defined over Q . Let L be a lattice in E Q . The zeta series associated to ϕ and L is defined by ζ L ( s ) = l Γ L ' [ det ( Q ( l ) ) ] - s , s C where L ' = { l L | det ( Q ( l ) ) 0 } ,...

Zhang-Zagier heights of perturbed polynomials

Christophe Doche (2001)

Journal de théorie des nombres de Bordeaux

In a previous article we studied the spectrum of the Zhang-Zagier height [2]. The progress we made stood on an algorithm that produced polynomials with a small height. In this paper we describe a new algorithm that provides even smaller heights. It allows us to find a limit point less than 1 . 289735 i.e. better than the previous one, namely 1 . 2916674 . After some definitions we detail the principle of the algorithm, the results it gives and the construction that leads to this new limit point.

Z-modules

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].

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