Probabilistic Number Theory I

Elliott, P. D. T. A.

doi:10.1007/978-1-4612-9989-9

Cited by 287 publications

(172 citation statements)

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“…Therefore the model is mostly adapted to the analysis of the distribution of the small divisors of an integer. See [3] p.122, see also [11] (Introduction) for a complete and precise analysis of this point.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Instants of small amplitude of brownian motion and application to the Kubilius model

Weber

2013

Period Math Hung

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Abstract. Let W (t), t ≥ 0 be standard Brownian motion. We study the size of the time intervals I which are admissible for the long range of slow increase, namely given a real z > 0,and we estimate their number of occurences. We obtain optimal results in terms of class test functions and, by means of the quantitative Borel-Cantelli lemma, a fine frequency result concerning their occurences. Using Sakhanenko's invariance principe to transfer the results to the Kubilius model, we derive applications to the prime number divisor function. We obtain refinements of some results recently proved by Ford and Tenenbaum in [4]. 2010

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Section: Auxiliary Resultsmentioning

confidence: 99%

Instants of small amplitude of brownian motion and application to the Kubilius model

Weber

2013

Period Math Hung

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“…Note that for a suitably chosen q, Applying the Hardy-Littlewood tauberian Theorem (Hardy (1949), Elliott, (1979, Chapter 2) or a method of Daboussi and Delange (see E Lemma 9) we obtain the convergence of the series 2 p~\g(p) -i).…”

Section: \G(p)-\\>l/2pmentioning

confidence: 99%

“…Otherwise T(«) = 0 would hold on a sequence of integers of asymptotic density one; almost always. (See, for example, Elliott (1979) …”

Section: T(p)-0pmentioning

confidence: 99%

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Multiplicative functions and Ramanujan's τ-function

Elliott

1981

J. Aust. Math. Soc.

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“…Next we ask whether there is a distribution function for ω(n)? In other words if, typically, the distance between ω(n) and log log n is roughly of size √ log log n can we say anything about the distribution of (4) ω(n) − log log n √ log log n ?…”

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confidence: 99%