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Landau's function for one million billions | | Marc Deleglise
; Jean-Louis Nicolas
; Paul Zimmermann
; | | Date: |
14 Mar 2008 | | Abstract: | 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 $M=q_1^{al_1}q_2^{al_2}... q_k^{al_k}$,
we define $ell(M)$ to be $q_1^{al_1}+q_2^{al_2}+... +q_k^{al_k}$; one
century ago, E. Landau proved that $g(n)=max_{ell(M)le n} M$ and that, when
$n$ goes to infinity, $log g(n) sim sqrt{nlog(n)}$. There exists a basic
algorithm to compute $g(n)$ for $1 le n le N$; its running time is
$co(N^{3/2}/sqrt{log N})$ 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 $ au(n)=sum_{ddv n} 1$. | | Source: | arXiv, 0803.2160 | | Services: | Forum | Review | PDF | Favorites | |
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