Explicit upper bounds for the remainder term in the divisor problem

Berkane, D.; Bordellès, O.; Ramaré, Olivier

Document type :

Article dans une revue scientifique: Article original

DOI :

10.1090/S0025-5718-2011-02535-4

Title :

Explicit upper bounds for the remainder term in the divisor problem

Author(s) :

Berkane, D. [Auteur]
Bordellès, O. [Auteur]
Ramaré, Olivier [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Centre National de la Recherche Scientifique [CNRS]

Journal title :

Mathematics of Computation

Pages :

1025-1051

Publisher :

American Mathematical Society

Publication date :

2012-03-01

ISSN :

0025-5718

HAL domain(s) :

Mathématiques [math]/Théorie des nombres [math.NT]

English abstract : [en]

We first report on computations made using the GP/PARI package that show that the error term ∆(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 ...
Show more >We first report on computations made using the GP/PARI package that show that the error term ∆(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 $|\Delta(x)| \le 0.397 x^{1/2}$ when $x \ge 5 560$ and that $|\Delta(x)| \le 0.764 x^{1/3}\log x$ 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 concerningShow less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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