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Erschienen in:
2006 | OriginalPaper | Buchkapitel
Estimates Involving Finite Groups and Semi-Simple Rings
Erschienen in: Handbook of Number Theory I
Verlag: Springer Netherlands
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Excerpt
Let o(σ) be the order of the element σ in the symmetric group S n and denote
. Then:
a)
Let ɛ>0. Then there exist n0(ɛ) such that for n≥n0(ɛ),
b)
E. Landau. Handbuch der Lehre von der Verteilung der Primzahlen. I. pp. 222–229, Teubner, Leipzig, 1909.
c)
For almost all n we have
O. Herrmann. Aufgabe 546. Elemente der Math. 23 (1968), 41–42.
d)
Let ɛ>0. Then for n≥n0(ɛ),
where
M. Szalay. On a theorem of Landau. (Hungarian.) Mat. Lapok 22 (1971), 317–321.
e)
and the maximum of 
is attained for n=1319766
and the maximum of is attained for n=1319766
J.-P. Massias. Majoration explicite de l’ordre maximum d’un élément du groupe symétrique. Ann. Fac. Sci. Toulouse Math. (5) 6 (1984) 269–281.
f)
(a>0, constant.)
(a>0, constant.)
J.-P. Massias, J.-L. Nicolas and G. Robin. Évaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique. Acta Arith. 50 (1988), 221–242. See also
J.-P. Massias, J.-L. Nicolas and G. Robin. Effective bounds for the maximal order of an element in the symmetric group. Math. Comp. 53 (1989), no. 188, 665–678.
Remark. In these papers appear also results on P(M(n)), where P(k) denotes the greatest prime factor of k.
g)
for n≥3; and
for n≥93898
for n≥3; and
(See J.-P. Massias, J.-L. Nicolas and G. Robin (1989).)