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Journal Articles Journal de Théorie des Nombres de Bordeaux Year : 2008

Landau's function for one million billions


Let ${\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\mathfrak S}_n$. If the standard factorization of $M$ into primes is $M=q_1^{\al_1}q_2^{\al_2}\ldots q_k^{\al_k}$, we define $\ell(M)$ to be $q_1^{\al_1}+q_2^{\al_2}+\ldots +q_k^{\al_k}$; one century ago, E. Landau proved that $g(n)=\max_{\ell(M)\le n} M$ and that, when $n$ goes to infinity, $\log g(n) \sim \sqrt{n\log(n)}$. There exists a basic algorithm to compute $g(n)$ for $1 \le n \le N$; its running time is $\co\left(N^{3/2}/\sqrt{\log N}\right)$ and the needed memory is $\co(N)$; 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 {\it $\ell$-superchampion numbers}. Similar numbers, the {\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function $\tau(n)=\sum_{d\dv n} 1$.
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Dates and versions

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



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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