Landau’s function for one million billions
Journal de Théorie des Nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 625-671.

Soit 𝔖 n le groupe symétrique sur n lettres et g(n) l’ordre maximal d’un élément de 𝔖 n . Si la factorisation en nombres premiers de M est M=q 1 α 1 q 2 α 2 ...q k α k , nous définissons (M) comme étant q 1 α 1 +q 2 α 2 +...+q k α k  ; il y a un siècle, E. Landau a montré que g(n)=max (M)n M et que, quand n tend vers l’infini, logg(n)nlog(n).

Il existe un algorithme élémentaire pour calculer g(n) pour 1nN ; son temps d’exécution est en 𝒪N 3/2 /logN et la place mémoire nécessaire est en 𝒪(N) ; cela permet de calculer g(n) jusqu’à, disons, un million. Nous donnons un algorithme pour calculer g(n) pour n jusqu’à 10 15 . L’idée principale est de considérer les nombres dits -superchampions. Des nombres similaires, les nombres hautement composés supérieurs, ont été introduits par S. Ramanujan pour étudier les grandes valeurs de la fonction nombre de diviseurs τ(n)= d|n 1.

Let 𝔖 n denote the symmetric group with n letters, and g(n) the maximal order of an element of 𝔖 n . If the standard factorization of M into primes is M=q 1 α 1 q 2 α 2 ...q k α k , we define (M) to be q 1 α 1 +q 2 α 2 +...+q k α k ; one century ago, E. Landau proved that g(n)=max (M)n M and that, when n goes to infinity, logg(n)nlog(n).

There exists a basic algorithm to compute g(n) for 1nN; its running time is 𝒪N 3/2 /logN and the needed memory is 𝒪(N); it allows computing g(n) up to, say, one million. We describe an algorithm to calculate g(n) for n up to 10 15 . The main idea is to use the so-called -superchampion numbers. Similar numbers, the superior highly composite numbers, were introduced by S. Ramanujan to study large values of the divisor function τ(n)= d|n 1.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.644
Mots clés : Arithmetical function, symmetric group, maximal order, highly composite number.
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     author = {Marc Del\'eglise and Jean-Louis Nicolas and Paul Zimmermann},
     title = {Landau{\textquoteright}s function for one million billions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {625--671},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {3},
     year = {2008},
     doi = {10.5802/jtnb.644},
     zbl = {pre05572695},
     mrnumber = {2523311},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.644/}
}
Marc Deléglise; Jean-Louis Nicolas; Paul Zimmermann. Landau’s function for one million billions. Journal de Théorie des Nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 625-671. doi : 10.5802/jtnb.644. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.644/

[1] E. Bach, Sums over Primes. Conference on Algorithmic Number Theory, Turku, May 8-11, 2007.

[2] E. Bach and J. Sorenson, Computing prime harmonic sums. Submitted to Math. Comp.

[3] M. Deléglise and J. Rivat, Computing π(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method. Math. Comp. 65 (1996), 235–245. | MR 1322888 | Zbl 0869.11068

[4] H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica 2 (1936), 23–46. | Zbl 0015.19702

[5] P. Dusart, The k th prime is greater than k(logk+loglogk-1) for k2. Math. Comp. 68 (1999), 411–415. | MR 1620223 | Zbl 0913.11039

[6] P. Erdős and P. Turán, On some problems of a statistical group theory, IV. Acta Math. Acad. Sci. Hungar. 19 (1968), 413–435. | MR 232833 | Zbl 0235.20004

[7] H. Gerlach, Über die Elemente einer Menge verallgemeinerter ganzer Zahlen, die klein sind bezüglich einer auf dieser Menge definierten reellwertigen Abbildung. Thesis of the University of Kaiserslautern, 1986. | Zbl 0606.10037

[8] J. Grantham, The largest prime dividing the maximal order of an element of S n . Math. Comp. 64 (1995), 407–410. | MR 1270619 | Zbl 0824.11059

[9] E. Landau, Über die Maximalordnung der Permutationen gegebenen Grades. Archiv. der Math. und Phys., Sér 3, 5 (1903), 92–103. Handbuch der Lehre von der Verteilung der Primzahlen, I, 2nd ed, Chelsea, New-York, 1953, 222–229.

[10] G. Levitt and J.-L. Nicolas, On the Maximum Order of Torsion Elements in GL(n,) and Aut(F n ). Journal of Algebra 208 (1998), 630–642. | MR 1655470 | Zbl 0930.11070

[11] J.-P. Massias, Majoration explicite de l’ordre maximum d’un élément du groupe symétrique. Ann. Fac. Sci. Toulouse Math. 6 (1984), 269–280. | Numdam | MR 799599 | Zbl 0574.10043

[12] J.-P. Massias, J.-L. Nicolas et G. Robin. Évaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique. Acta Arithmetica 50 (1988), 221–242. | MR 960551 | Zbl 0588.10049

[13] 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), 665–678. | MR 979940 | Zbl 0675.10028

[14] W. Miller, The Maximal Order of an Element of a Finite Symmetric Group. Amer. Math. Monthly 94 (1987), 497–506. | MR 935414

[15] F. Morain, Table de g(n) pour 1n32000. Internal document, INRIA, 1988.

[16] T. R. Nicely, http://www.trnicely.net/gaps/gaplist.html

[17] J.-L. Nicolas, Sur l’ordre maximum d’un élément dans le groupe S n des permutations. Acta Arithmetica 14 (1968), 315–332. | MR 230803 | Zbl 0179.34804

[18] J.-L. Nicolas, Ordre maximal d’un élément du groupe des permutations et highly composite numbers. Bull. Soc. Math. France 97 (1969), 129–191. | Numdam | MR 254130 | Zbl 0184.07202

[19] J.-L. Nicolas, Calcul de l’ordre maximum d’un élément du groupe symétrique S n . Rev. Française Informat. Recherche Opérationnelle, Sér. R-2 3 (1969), 43–50. | Numdam | MR 253514 | Zbl 0186.30202

[20] J.-L. Nicolas, On highly composite numbers. Ramanujan revisited, edited by G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, R. A. Rankin, Academic Press, 1988, 216–244. | MR 938967 | Zbl 0651.10029

[21] J.-L. Nicolas, On Landau’s function g(n). The Mathematics of Paul Erdős, vol. I, R. L. Graham and J. Nešetřil editors, Springer Verlag, Algorithms and Combinatorics no 13 (1997), 228–240. | MR 1425188 | Zbl 0869.11077

[22] J.-L. Nicolas, Comparaison des ordres maximaux dans les groupes S n et GL(n,). Acta Arithmetica 96 (2000), 175–203. | MR 1814452 | Zbl 1016.11043

[23] J.-L. Nicolas and N. Zakic, Champion numbers for the number of representations as a sum of six squares. In preparation.

[24] S. Ramanujan, Highly composite numbers. Proc. London Math. Soc. Serie 2, 14 (1915), 347–409. Collected papers, Cambridge University Press, 1927, 78–128. | MR 2280858

[25] S. Ramanujan, Highly composite numbers, annotated by J.-L. Nicolas and G. Robin. The Ramanujan J. 1 (1997), 119–153. | MR 1606180 | Zbl 0917.11043

[26] H. Riesel, Prime Numbers and Computer Methods for Factorization. Birkhäuser, 1985. | MR 897531 | Zbl 0582.10001

[27] G. Robin, Méthodes d’optimisation pour un problème de théorie des nombres. R.A.I.R.O. Informatique Théorique 17 (1983), 239–247. | Numdam | MR 743888 | Zbl 0531.10012

[28] J. B. Rosser and L. Schoenfeld, Approximate Formulas for Some Functions of Prime Numbers. Illinois. J. Math 6 (1962), 64–94. | MR 137689 | Zbl 0122.05001

[29] S. M. Shah, An inequality for the arithmetical function g(x). J. Indian Math. Soc. 3 (1939), 316–318. | MR 1236

[30] M. Szalay, On the maximal order in S n and S n * . Acta Arithmetica 37 (1980), 321–331. | MR 598884 | Zbl 0443.10029

[31] P. M. B. Vitányi, On the size of DOL languages. Lecture Notes in Computer Science, vol. 15, Springer-Verlag, 1974, 78–92 and 327–338. | MR 423893 | Zbl 0293.68062