Fonctions arithmétiques et formes binaires irréductibles de degré 3
Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1297-1363.

Dans cet article, nous obtenons des estimations de l’ordre moyen, sur les valeurs de la forme cubique X 1 3 +2X 2 3 , de fonctions multiplicatives h soumises à certaines conditions. On donne en particulier une formule asymptotique du nombre d’entiers friables de la forme n 1 3 +2n 2 2 , valide pour un paramètre de friabilité non borné. La méthode utilisée s’applique également à des fonctions multiplicatives oscillantes comme la fonction μ de Mœbius : il s’ensuit une nouvelle preuve de la conjecture de Chowla pour la forme X 1 3 +2X 2 3 , récemment démontrée par Helfgott dans le cas plus général des formes binaires cubiques irréductibles.

In this article, we give some estimates for the average order, over the values of the cubic form X 1 3 +2X 2 3 , for some multiplicative functions h satisfying certain conditions. We provide an asymptotic formula for the number of y-friable values of n 1 3 +2n 2 3 , valid in an unbounded range. Our method also applies to some oscillating multiplicative functions like the Mœbius function μ : this gives another proof of the Chowla conjecture for the form X 1 3 +2X 2 3 recently proved by Helfgott in the more general case of binary and irreducible cubic forms.

Reçu le : 2016-03-06
Révisé le : 2016-05-25
Accepté le : 2017-07-12
Publié le : 2018-05-03
DOI : https://doi.org/10.5802/aif.3189
Classification : 11E76,  11N25,  11N36,  11N37,  11Y05
Mots clés : Entiers friables, fonctions multiplicatives, cribles, formes binaires
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     author = {Lachand, Armand},
     title = {Fonctions arithm\'etiques et formes binaires irr\'eductibles de degr\'e <span class="mathjax-formula">$3$</span>},
     journal = {Annales de l'Institut Fourier},
     pages = {1297--1363},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {3},
     year = {2018},
     doi = {10.5802/aif.3189},
     language = {fr},
     url = {archive.numdam.org/item/AIF_2018__68_3_1297_0/}
}
Lachand, Armand. Fonctions arithmétiques et formes binaires irréductibles de degré $3$. Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1297-1363. doi : 10.5802/aif.3189. http://archive.numdam.org/item/AIF_2018__68_3_1297_0/

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