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Explicit upper bounds for the remainder term in the divisor problem

Authors: D. Berkane, O. Bordellès and O. Ramaré
Journal: Math. Comp. 81 (2012), 1025-1051
MSC (2010): Primary 11N56; Secondary 11N37
Published electronically: August 25, 2011
MathSciNet review: 2869048
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We first report on computations made using the GP/PARI package that show that the error term $ \Delta(x)$ in the divisor problem is $ =\mathscr{M}(x,4)+ O^*(0.35 x^{1/4}\log x)$ when $ x$ ranges $ [1 081 080, 10^{10}]$, where $ \mathscr{M}(x,4)$ is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that $ \vert\Delta(x)\vert\le 0.397 {x^{1/2}}$ when $ x\ge 5 560$ and that $ \vert\Delta(x)\vert\le 0.764 {x^{1/3}\log x}$ when $ x\ge 9 995$. Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning $ \Delta(x)$.

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Additional Information

D. Berkane
Affiliation: Département de Mathématiques, Université de Blida, 270 route de soumaa, 09 000 Blida, Algérie

O. Bordellès
Affiliation: 2, allée de la combe, 43 000 Aiguilhe, France

O. Ramaré
Affiliation: CNRS / Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq cedex, France

Received by editor(s): January 3, 2011
Received by editor(s) in revised form: February 16, 2011
Published electronically: August 25, 2011
Dedicated: To the memory of John Selfridge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.