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Person# Nicolas Bersier

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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. [9], 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.