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Effective bounds for the maximal order of an element in the symmetric group. (English) Zbl 0675.10028

Let g(n) denote the largest order of an element of the symmetric group \(S_ n\). The authors recently [Acta Arith. 50, No.3, 221-242 (1988; Zbl 0588.10049)] showed that, as \(n\to \infty\), \[ \log g(n)=\sqrt{li^{- 1}(n)}+O(\sqrt{n} \exp (-c\sqrt{\log n})), \] where li(x) is the logarithmic integral and c is a positive constant. In the present paper the authors prove explicit, though asymptotically weaker, upper and lower bounds for log g(n) in terms of the quantity \[ (*)\quad \sqrt{n \log n}(1+\frac{\log \log n-a}{2 \log n}) \] for various values of a. For example, it is shown that log g(n) is bounded from above by (*) with \(a=0.975\) for all \(n\geq 3\), and bounded from below by (*) with \(a=1.2\) for all \(n\geq 93,898\).
Reviewer: A.Hildebrand

MSC:

11N37 Asymptotic results on arithmetic functions
11N45 Asymptotic results on counting functions for algebraic and topological structures
11-04 Software, source code, etc. for problems pertaining to number theory
20B30 Symmetric groups
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References:

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