Integers free of large prime factors and the Riemann hypothesis

Hildebrand, Adolf

doi:10.1112/s0025579300012481

Cited by 41 publications

(39 citation statements)

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“…The range of y such that the asymptotic formula (x, y) ∼ xρ(u), where x = y u , has been significantly enlarged by De Bruijn [6][7][8] (y ≥ exp((ln x) 5/8+ε )) and Hildebrand [22] (y ≥ exp((ln ln x) 5/3+ε )). Notice that in our ensemble m (where each element is weighted, not simply counted) we have x = p 1 p 2 · · · p m and y = p m and thus y ∼ ln x.…”

Section: On the Dickman-de Bruijn Distributionmentioning

confidence: 99%

The Möbius function and statistical mechanics

Cellarosi

Sinaĭ

2011

Bull. Math. Sci.

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We consider a probabilistic model for square-free numbers, and provide limit theorems for several random variables defined in our ensemble. The limit transition corresponds to the thermodynamical limit in Statistical Mechanics. We also prove some inequalities inspired by a recent conjecture by P. Sarnak concerning the randomness in the Möbius sequence, and discuss a method of summation for the Riemann zeta function ζ(s) on the vertical line s = 1.

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Section: On the Dickman-de Bruijn Distributionmentioning

confidence: 99%

The Möbius function and statistical mechanics

Cellarosi

Sinaĭ

2011

Bull. Math. Sci.

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“…Par exemple, Hildebrand [8] a montré que l'hypothèse de Riemann est équivalente à l'assertion que pour tout ε > 0, on a…”

Section: Théorème De Bombieri-vinogradov Pondéré Pour Les Entiers Friunclassified

“…Quitte à augmenter la valeur de c pour avoir x 3/4 ≤ Ψ(x, y)/(log x) 8 , la formule (18) avec Q = x 1/3 et les calculs de [4] (plus précisément la formule précédant la formule (7.6)) montrent que le membre de droite de (19) est…”

Section: Théorème D'erdös-kác Sur Les Friables Translatésunclassified

Propriétés multiplicatives des entiers friables translatés

Drappeau

2014

Colloq. Math.

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Abstract. An integer is said to be y-friable if its greatest prime factor P (n) is less than y. In this paper, we study numbers of the shape n − 1 when P (n) ≤ y and n ≤ x. One expects that, statistically, their multiplicative behaviour resembles that of all integers less than x. Extending a result of Basquin [1], we estimate the mean value over shifted friable numbers of certain arithmetic functions when (log x) c ≤ y for some positive c, showing a change in behaviour according to whether log y/ log log x tends to infinity or not. In the same range in (x, y), we prove an Erdös-Kac-type theorem for shifted friable numbers, improving a result of Fouvry & Tenenbaum [4]. The results presented here are obtained using recent work of Harper [6] on the statistical distribution of friable numbers in arithmetic progressions.

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“…Lorsque δ 1, nos conditions de validité sont notablement moins restrictives que (1·6) et coïncident avec celles de la formule de Hildebrand [20] si f = 1 ou celles de Smida [37] si f = τ k : elles sont donc essentiellement optimales en l'état actuel de nos connaissances sur les zéros de la fonction zêta de Riemann -cf. Hildebrand [19].…”

Section: éNoncé Des Résultatsunclassified