On the parity of generalized partition functions, III

Fethi Ben Saïd; Jean-Louis Nicolas; Ahlem Zekraoui

doi:10.5802/jtnb.704

On the parity of generalized partition functions, III

Fethi Ben Saïd ¹ ; Jean-Louis Nicolas ² ; Ahlem Zekraoui ¹

¹ Université de Monastir Faculté des Sciences de Monastir Avenue de l’Environement 5000 Monastir, Tunisie
² Université de Lyon 1 Institut Camile Jordan, UMR 5208 Batiment Doyen Jean Braconnier 21 Avenue Claude Bernard F-69622 Villeurbanne, France

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 51-78.

Résumé
Abstract

Dans cet article, nous complétons les résultats de J.-L. Nicolas [15], en déterminant tous les éléments de l’ensemble $𝒜 = 𝒜 (1 + z + z^{3} + z^{4} + z^{5})$ pour lequel la fonction de partition $p (𝒜, n)$ (c-à-d le nombre de partitions de $n$ en parts dans $𝒜$ ) est paire pour tout $n \geq 6$ . Nous donnons aussi un équivalent asymptotique à la fonction de décompte de cet ensemble.

Improving on some results of J.-L. Nicolas [15], the elements of the set $𝒜 = 𝒜 (1 + z + z^{3} + z^{4} + z^{5})$ , for which the partition function $p (𝒜, n)$ (i.e. the number of partitions of $n$ with parts in $𝒜$ ) is even for all $n \geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

MR Zbl

DOI : 10.5802/jtnb.704

Classification : 11P81, 11N25, 11N37
Mots clés : Partitions, periodic sequences, order of a polynomial, orbits, $2$-adic numbers, counting function, Selberg-Delange formula.

Affiliations des auteurs :

Fethi Ben Saïd ¹ ; Jean-Louis Nicolas ² ; Ahlem Zekraoui ¹

¹ Université de Monastir Faculté des Sciences de Monastir Avenue de l’Environement 5000 Monastir, Tunisie
² Université de Lyon 1 Institut Camile Jordan, UMR 5208 Batiment Doyen Jean Braconnier 21 Avenue Claude Bernard F-69622 Villeurbanne, France

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {51--78},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
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     year = {2010},
     doi = {10.5802/jtnb.704},
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     zbl = {1236.11088},
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Fethi Ben Saïd; Jean-Louis Nicolas; Ahlem Zekraoui. On the parity of generalized partition functions, III. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 51-78. doi : 10.5802/jtnb.704. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/

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