In this article, we give some estimates for the average order, over the values of the cubic form , for some multiplicative functions satisfying certain conditions. We provide an asymptotic formula for the number of -friable values of , valid in an unbounded range. Our method also applies to some oscillating multiplicative functions like the Mœbius function : this gives another proof of the Chowla conjecture for the form recently proved by Helfgott in the more general case of binary and irreducible cubic forms.
Dans cet article, nous obtenons des estimations de l’ordre moyen, sur les valeurs de la forme cubique , de fonctions multiplicatives soumises à certaines conditions. On donne en particulier une formule asymptotique du nombre d’entiers friables de la forme , valide pour un paramètre de friabilité non borné. La méthode utilisée s’applique également à des fonctions multiplicatives oscillantes comme la fonction de Mœbius : il s’ensuit une nouvelle preuve de la conjecture de Chowla pour la forme , récemment démontrée par Helfgott dans le cas plus général des formes binaires cubiques irréductibles.
Revised:
Accepted:
Published online:
Mot clés : Entiers friables, fonctions multiplicatives, cribles, formes binaires
Keywords: Friable integers, multiplicative functions, sieves, binary forms
@article{AIF_2018__68_3_1297_0, author = {Lachand, Armand}, title = {Fonctions arithm\'etiques et formes binaires irr\'eductibles de degr\'e $3$}, journal = {Annales de l'Institut Fourier}, pages = {1297--1363}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {3}, year = {2018}, doi = {10.5802/aif.3189}, language = {fr}, url = {http://archive.numdam.org/articles/10.5802/aif.3189/} }
TY - JOUR AU - Lachand, Armand TI - Fonctions arithmétiques et formes binaires irréductibles de degré $3$ JO - Annales de l'Institut Fourier PY - 2018 SP - 1297 EP - 1363 VL - 68 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3189/ DO - 10.5802/aif.3189 LA - fr ID - AIF_2018__68_3_1297_0 ER -
%0 Journal Article %A Lachand, Armand %T Fonctions arithmétiques et formes binaires irréductibles de degré $3$ %J Annales de l'Institut Fourier %D 2018 %P 1297-1363 %V 68 %N 3 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3189/ %R 10.5802/aif.3189 %G fr %F AIF_2018__68_3_1297_0
Lachand, Armand. Fonctions arithmétiques et formes binaires irréductibles de degré $3$. Annales de l'Institut Fourier, Volume 68 (2018) no. 3, pp. 1297-1363. doi : 10.5802/aif.3189. http://archive.numdam.org/articles/10.5802/aif.3189/
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