Le carré de la fonction des diviseurs
Let d(n) denote Dirichlet's divisor function for positive integer numbers. This work is primarily concerned with the study of We are interested, in the error term where Ρ3 is a polynomial of degree 3 ; more precisely xΡ3(log x) is the residue of in s = 1. A. Ivić showed that E(x) = Ο(x1/2+ε) for all ε > 0 (cf. , p.394). We will prove that for all x > 0, we have With this intention, we apply Perron's formula to the generating function ζ4(s)/ ζ (2s) and Landau's finite difference method. It was conjectured that E(x) = Ο(x1/4+ε) for ε > 0. The existence of non-trivial zeros of the Riemann ζ function implies that we cannot do better, that is The study of the Riesz means for ρ sufficiently large shows that their error term, is an infinite series , on the zeros of the Riemann Zeta function added with a development , into a series of Hardy-Voronoï's type, both being convergent. To find the "meaning" of , one could consider the difference But the series (probably) doesn't converge.We will thus substract only a finite part of , weighted by a smooth function ω, the number of terms of the finite part depending on x. If we consider this new error term , we obtain, using a classical method due to Hardy, that for x ≥ 1.
Keywords: Analytic number theory ; asymptotical analysis ; summatory functions ; generating functions ; Dirichlet series ; finite differences ; Théorie analytique des nombres ; analyse asymptotique ; fonctions sommatoires ; fonctions génératrices ; séries de Dirichlet ; différences finiesThèse École polytechnique fédérale de Lausanne EPFL, n° 3940 (2007)
Programme doctoral Mathématiques
Faculté des sciences de base
Institut d'analyse et calcul scientifique
Chaire de théorie des nombres
Record created on 2007-09-03, modified on 2016-08-08