Landau's function
In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn. Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence. For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6. An element of order 6 in the group S5 can be written in cycle notation as (1 2) (3 4 5). Note that the same argument applies to the number 6, that is, g(6) = 6. There are arbitrarily long sequences of consecutive numbers n, n + 1, ..., n + m on which the function g is constant.[1]
The integer sequence g(0) = 1, g(1) = 1, g(2) = 2, g(3) = 3, g(4) = 4, g(5) = 6, g(6) = 6, g(7) = 12, g(8) = 15, ... (sequence A000793 in the OEIS) is named after Edmund Landau, who proved in 1902[2] that
- [math]\displaystyle{ \lim_{n\to\infty}\frac{\ln(g(n))}{\sqrt{n \ln(n)}} = 1 }[/math]
(where ln denotes the natural logarithm). Equivalently (using little-o notation), [math]\displaystyle{ g(n) = e^{(1+o(1))\sqrt{n\ln n}} }[/math].
More precisely,[3]
- [math]\displaystyle{ \ln g(n)=\sqrt{n\ln n}\left(1+\frac{\ln\ln n-1}{2\ln n}-\frac{(\ln\ln n)^2-6\ln\ln n+9}{8(\ln n)^2}+O\left(\left(\frac{\ln\ln n}{\ln n}\right)^3\right)\right). }[/math]
If [math]\displaystyle{ \pi(x)-\operatorname{Li}(x)=O(R(x)) }[/math], where [math]\displaystyle{ \pi }[/math] denotes the prime counting function, [math]\displaystyle{ \operatorname{Li} }[/math] the logarithmic integral function with inverse [math]\displaystyle{ \operatorname{Li}^{-1} }[/math], and we may take [math]\displaystyle{ R(x)=x\exp\bigl(-c(\ln x)^{3/5}(\ln\ln x)^{-1/5}\bigr) }[/math] for some constant c > 0 by Ford,[4] then[3]
- [math]\displaystyle{ \ln g(n)=\sqrt{\operatorname{Li}^{-1}(n)}+O\bigl(R(\sqrt{n\ln n})\ln n\bigr). }[/math]
The statement that
- [math]\displaystyle{ \ln g(n)\lt \sqrt{\mathrm{Li}^{-1}(n)} }[/math]
for all sufficiently large n is equivalent to the Riemann hypothesis.
It can be shown that
- [math]\displaystyle{ g(n)\le e^{n/e} }[/math]
with the only equality between the functions at n = 0, and indeed
- [math]\displaystyle{ g(n) \le \exp\left(1.05314\sqrt{n\ln n}\right). }[/math][5]
Notes
- ↑ Nicolas, Jean-Louis (1968), "Sur l’ordre maximum d’un élément dans le groupe Sn des permutations" (in French), Acta Arithmetica 14: 315–332
- ↑ Landau, pp. 92–103
- ↑ 3.0 3.1 Massias, J. P.; Nicholas, J. L.; Robin, G. (1988), "Évaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique" (in French), Acta Arithmetica 50: 221–242
- ↑ Kevin Ford (November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function". Proc. London Math. Soc. 85 (3): 565–633. doi:10.1112/S0024611502013655. https://faculty.math.illinois.edu/~ford/wwwpapers/zetabd.pdf.
- ↑ Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, pp. 269–281 (1985).
References
- E. Landau, "Über die Maximalordnung der Permutationen gegebenen Grades [On the maximal order of permutations of given degree]", Arch. Math. Phys. Ser. 3, vol. 5, 1903.
- W. Miller, "The maximum order of an element of a finite symmetric group", American Mathematical Monthly, vol. 94, 1987, pp. 497–506.
- J.-L. Nicolas, "On Landau's function g(n)", in The Mathematics of Paul Erdős, vol. 1, Springer-Verlag, 1997, pp. 228–240.
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