Phénomène de Moser-Newman pour les nombres sans facteur carré

Mauduit, Christian; Moreira, Carlos Gustavo

doi:10.24033/bsmf.2699

Mauduit, Christian ; Moreira, Carlos Gustavo

Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 3, pp. 599-617

Résumé

Let $μ$ denotes the Möbius function and $𝐭 = {(t (n))}_{n \in ℕ}$ be the Thue-Morse sequence, defined by $t (n) = + 1$ if the number of 1 in the dyadic representation of $n$ is even and $t (n) = - 1$ otherwise. The aim of this work is to give an asymptotic formula for $S (N) = \sum_{n < N} μ^{2} (n) t (n)$ and to prove that $S (N) < 0$ for $N$ big enough. This shows that square-free integers provide a first example of non-linear Moser-Newman phenomenon. Our method gives a similar result for $z$ -th power free integers.

Publié le : 2015-07-21

MR Zbl

DOI : 10.24033/bsmf.2699

Classification : 11A63, 11B85, 11L03, 11N25
Mots-clés : Suite de Thue-Morse, nombres sans facteur carré, sommes trigonométriques.

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     author = {Mauduit, Christian and Moreira, Carlos Gustavo},
     title = {Ph\'enom\`ene de {Moser-Newman} pour les nombres sans facteur carr\'e},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {599--617},
     year = {2015},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {143},
     number = {3},
     doi = {10.24033/bsmf.2699},
     mrnumber = {3417734},
     zbl = {1352.11015},
     language = {fr},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2699/}
}

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Mauduit, Christian; Moreira, Carlos Gustavo. Phénomène de Moser-Newman pour les nombres sans facteur carré. Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 3, pp. 599-617. doi: 10.24033/bsmf.2699

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