On the number of prime factors of an integer

Hildebrand, Adolf; Tenenbaum, Gérald

doi:10.1215/s0012-7094-88-05620-7

Cited by 45 publications

(37 citation statements)

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“…The asymptotic behaviour of N 16 (x) is dominated by the number of prime pairs (p, q) with p < q such that r 2 (p) = r 2 (q) = 2, (ord 2 (p), ord 2 (q)) = 1 and p · q ≤ 2x − 1. We are inclined to believe that this number is a positive fraction of all pairs (p, q) with p < q and such that p · q ≤ 2x − 1, which, as is well-known [8], grows asymptotically as 2x log log x/ log x. Thus we are tempted to conjecture that N 16 (x) ∼ c 0 x log log x log x , for some positive constant c 0 .…”

Section: · · · Es Lsmentioning

confidence: 97%

“…Since e 1 ≡ 2(mod 4), it follows by Theorem 3 that there are infinitely many primes q satisfying (8).…”

Section: Proposition 4 (Grh)mentioning

confidence: 99%

“…The celebrated [24,12,8] Golay code arises on taking n = 12. More generally, we obtain duadic codes with multiplier −1 which have received much attention in the last twenty years [14,27].…”

Section: Codes and Pelikán's Conjecturementioning

confidence: 99%

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Around Pelikán’s conjecture on very odd sequences

Moree

Solé²

2005

manuscripta math.

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Very odd sequences were introduced in 1973 by Pelikán who conjectured that there were none of length ≥ 5. This conjecture was disproved first by MacWilliams and Odlyzko [17] in 1977 and then by two different sets of authors in 1992 [1], 1995 [9]. We give connections with duadic codes, cyclic difference sets, levels (Stufen) of cyclotomic fields, and derive some new asymptotic results on the length of very odd sequences and the number of such sequences of a given length.

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Section: · · · Es Lsmentioning

confidence: 97%

“…Since e 1 ≡ 2(mod 4), it follows by Theorem 3 that there are infinitely many primes q satisfying (8).…”

Section: Proposition 4 (Grh)mentioning

confidence: 99%

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Around Pelikán’s conjecture on very odd sequences

Moree

Solé²

2005

manuscripta math.

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“…D'autre part, grâce à une méthode nouvelle fondée sur la méthode du col, Hildebrand et Tenenbaum ont démontré dans [7] la relation n(x,k+l)_L( /logL\\ (2) n(…”

Section: P\ Riunclassified

“…Nous laisserons ouverte cette importante question. Notre approche, indépen-dante des méthodes et des résultats de l'article [7] de Hildebrand et …”

Section: P\ Riunclassified

Unimodalité de la distribution du nombre de diviseurs premiers d'un entier

Balazard¹

1990

Annales De L’institut Fourier

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A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Hwang

1998

Random Struct. Alg.

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Let F X denote a polynomial ring over a finite field F with q elements. q qLet P P be the set of monic polynomials over F of degree n. Assuming that each of the q n n q possible monic polynomials in P P is equally likely, we give a complete characterization of n Ž . the limiting behavior of P ⍀ s m as n ª ϱ by a uniform asymptotic formula valid for n Ž . m G 1 and n y m ª ϱ, where ⍀ represents the number multiplicities counted of irren ducible factors in the factorization of a random polynomial in P P . The distribution of ⍀ is n n essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q y1 . Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional Ž behavior being essentially a parabolic cylinder function or some linear combinations of the . standard normal law and its iterated integrals . As applications of this uniform asymptotic Ž . formula, we derive most known results concerning P ⍀ s m and present many new ones n like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Renyi's problem, concerning thé Ž . distribution of the difference of the total number of irreducibles and the number of distinct irreducibles, is also presented.

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On the number of prime factors of an integer

Cited by 45 publications

References 4 publications

Around Pelikán’s conjecture on very odd sequences

Around Pelikán’s conjecture on very odd sequences

Unimodalité de la distribution du nombre de diviseurs premiers d'un entier

A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

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