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Sur les entiers inférieurs à x ayant plus de log ( x ) diviseurs

Marc DelégliseJean-Louis Nicolas — 1994

Journal de théorie des nombres de Bordeaux

Let τ ( n ) be the number of divisors of n ; let us define S λ ( x ) = C a r d n x ; τ ( n ) ( log x ) λ log 2 if λ 1 C a r d n x ; τ ( n ) < ( log x ) λ log 2 if λ < 1 It has been shown that, if we set f ( λ , x ) = x ( log x ) λ log λ - λ + 1 log log x the quotient S λ ( x ) / f ( λ , x ) is bounded for λ fixed. The aim of this paper is to give an explicit value for the inferior and superior limits of this quotient when λ 2 . For instance, when λ = 1 / log 2 , we prove lim inf S λ ( x ) f ( λ , x ) = 0 . 938278681143 and lim inf S λ ( x ) f ( λ , x ) = 1 . 148126773469

Landau’s function for one million billions

Marc DelégliseJean-Louis NicolasPaul Zimmermann — 2008

Journal de Théorie des Nombres de Bordeaux

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, log g ( n ) n log ( n ) . There exists a basic algorithm to compute g ( n ) for 1 n N ; its running time is 𝒪 N 3 / 2 / log N 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 ...

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