Let be the set of all nonnegative integers. For any positive integer and any subset of nonnegative integers, let be the number of solutions to the equation . In 2016, Qu proved that
providing that for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu’s result and obtained that
In this note, we further improve the lower bound on by showing that
Our bound reflects the correct order of magnitude of the representation function under the above restrictions due to the trivial fact that
Soit l’ensemble de tous les entiers non négatifs. Pour tout entier positif et tout sous-ensemble d’entiers non négatifs, notons le nombre de solutions de l’équation . En 2016, Qu a prouvé que
ce qui signifie que pour tous les entiers suffisamment grands, ce qui répondait par l’affirmative à un problème de Yang et Chen datant de 2012. Dans un article très récent, un autre Chen (le premier auteur dans notre article) a légèrement amélioré le résultat de Qu et obtenu que
Dans cette note, nous améliorons encore le minorant de en montrant que
Notre limite reflète l’ordre de grandeur correct de la fonction de représentation sous les restrictions ci-dessus en raison du fait trivial que .
Revised:
Accepted:
Published online:
Keywords: representation functions, order of functions, partitions of integers
Mots-clés : fonctions de représentation, ordre des fonctions, partitions d’entiers
Shi-Qiang Chen 1; Yuchen Ding 2; Xiaodong Lü 2; Yuhan Zhang 2

@article{CRMATH_2024__362_G5_547_0, author = {Shi-Qiang Chen and Yuchen Ding and Xiaodong L\"u and Yuhan Zhang}, title = {Correct order on some certain weighted representation functions}, journal = {Comptes Rendus. Math\'ematique}, pages = {547--552}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.573}, language = {en}, }
TY - JOUR AU - Shi-Qiang Chen AU - Yuchen Ding AU - Xiaodong Lü AU - Yuhan Zhang TI - Correct order on some certain weighted representation functions JO - Comptes Rendus. Mathématique PY - 2024 SP - 547 EP - 552 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.573 LA - en ID - CRMATH_2024__362_G5_547_0 ER -
Shi-Qiang Chen; Yuchen Ding; Xiaodong Lü; Yuhan Zhang. Correct order on some certain weighted representation functions. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 547-552. doi : 10.5802/crmath.573. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.573/
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