Number Theory/Algebra
Cesàro asymptotics for the orders of and as n→∞
Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 149-152.

Given an integer k>0, our main result states that the sequence of orders of the groups (respectively, of the groups ) is Cesàro equivalent as n→∞ to the sequence C1(k)nk2−1 (respectively, C2(k)nk2), where the coefficients C1(k) and C2(k) depend only on k; we give explicit formulas for C1(k) and C2(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ϕ(n) is Cesàro equivalent to . We present some experimental facts related to the main result.

Fixons un entier k>0. Notre resultat principal dit que la suite des ordres des groupes (respectivement, des groupes ) est equivalente au sens de Cesàro quand n→∞ à la suite C1(k)nk2−1 (respectivement, C2(k)nk2), où les coefficients C1(k) et C2(k) ne dependent que de k ; on donne des formules explicites pour C1(k) et C2(k). Ce resultat généralise le théorème (publié pour la première fois par I. Schoenberg) disant que la fonction d'Euler ϕ(n) est equivalente au sens de Cesàro à . On présente quelques faits experimentaux liés au resultat principal.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00328-5

Alexey G. Gorinov 1; Sergey V. Shadchin 2

1 Université Paris 7, U.F.R. de mathématiques, 2, place Jussieu, 75251, France
2 IHES, Bures-sur-Yvette, route de Chartres, 91140, France
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Alexey G. Gorinov; Sergey V. Shadchin. Cesàro asymptotics for the orders of  and  as n→∞. Comptes Rendus. Mathématique, Volume 337 (2003) no. 3, pp. 149-152. doi : 10.1016/S1631-073X(03)00328-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00328-5/`

[1] I. Schoenberg Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., Volume 28 (1928), pp. 171-199

[2] B.A. Venkov On a certain monotonic function, Uchenye Zapiski Leningrad State Univ. Math. Ser., Volume 111 (1949) no. 16, pp. 3-19 (in Russian)

[3] M. Kac; E.R. van Kampen; A. Wintner Ramanujan sums and almost periodic functions, Amer. J. Math., Volume 62 (1940), pp. 107-114

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