Notes on the domain of exponent pairs
Résumé
The theory of exponent pairs as initiated by Phillipps in 1933 proposes pairs of exponents pκ, λq so that one has ř n"N e 2iπφpnq !ε F κ`ε N λ`ε , for any positive ε, where φ is a 'monomial-like' smooth function whose first derivative is of size about F. We propose to explore the domain of available pairs pκ, λq through a very geometrical approach. We prove in particular that this domain is the convex hull of a connected curve in the classical case. We also show that a possible choice for λ, for any κ P r0, 1{2s, is given by λ " 1 ´κ log 2 log 2κ`1 2κ. We finally recall rapidly how this theory has been adapted to the higher dimensional setting. In passing, we take the opportunity of this slow-paced paper to describe some usage of the SageMath software.
Domaines
Théorie des nombres [math.NT]
Origine : Fichiers produits par l'(les) auteur(s)