Landau's function for one million billions (original) (raw)

Résumé

Let mathfrakSn{\mathfrak S}_nmathfrakSn denote the symmetric group with nnn letters, and g(n)g(n)g(n) the maximal order of an element of mathfrakSn{\mathfrak S}_nmathfrakSn. If the standard factorization of MMM into primes is M=q1al1q2al2ldotsqkalkM=q_1^{\al_1}q_2^{\al_2}\ldots q_k^{\al_k}M=q1al1q2al2ldotsqkalk, we define ell(M)\ell(M)ell(M) to be q1al1+q2al2+ldots+qkalkq_1^{\al_1}+q_2^{\al_2}+\ldots +q_k^{\al_k}q1al1+q2al2+ldots+qkalk; one century ago, E. Landau proved that g(n)=maxell(M)lenMg(n)=\max_{\ell(M)\le n} Mg(n)=maxell(M)lenM and that, when nnn goes to infinity, logg(n)simsqrtnlog(n)\log g(n) \sim \sqrt{n\log(n)}logg(n)simsqrtnlog(n). There exists a basic algorithm to compute g(n)g(n)g(n) for 1lenleN1 \le n \le N1lenleN; its running time is coleft(N3/2/sqrtlogNright)\co\left(N^{3/2}/\sqrt{\log N}\right)coleft(N3/2/sqrtlogNright) and the needed memory is co(N)\co(N)co(N); it allows computing g(n)g(n)g(n) up to, say, one million. We describe an algorithm to calculate g(n)g(n)g(n) for nnn up to 101510^{15}1015. The main idea is to use the so-called {\it ell\ellell-superchampion numbers}. Similar numbers, the {\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function tau(n)=sumddvn1\tau(n)=\sum_{d\dv n} 1tau(n)=sumddvn1.

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https://hal.science/hal-00264057

Soumis le : vendredi 14 mars 2008-09:49:56

Dernière modification le : mercredi 18 mars 2026-11:52:02

Archivage à long terme le : jeudi 20 mai 2010-22:04:07

Dates et versions

hal-00264057 , version 1 (14-03-2008)

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Marc Deléglise, Jean-Louis Nicolas, Paul Zimmermann. Landau's function for one million billions. Journal de Théorie des Nombres de Bordeaux, 2008, 20 (3), pp.625-671. ⟨10.5802/jtnb.644⟩. ⟨hal-00264057⟩

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