Another note on smooth numbers in short intervals

Matomäki, Kaisa

doi:10.1142/s1793042116500196

Cited by 2 publications

(4 citation statements)

References 10 publications

(16 reference statements)

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“…The later conjecture is motivated by attempts at rigorously estimating the running time of Lenstra's elliptic curve factoring algorithm [24,Section 6]. Our result also improves on earlier work of Croot [4], Matomäki [27,28] and Balog [1]. Finally for small fixed ε, a more difficult to state variant of Theorem 2 (see section 2) shows that C(ε) = ρ(1/ε) −13 is admissible, where ρ(u) is the Dickman-de Brujin function.…”

Section: Introductionsupporting

confidence: 76%

“…This improves on earlier work of Matomäki [28] and unpublished work of Hafner [13]. It would be interesting, in view of applications towards the complexity of Lenstra's elliptic curve factoring algorithm, to extend Corollary 6 to significantly smoother numbers (and one would naturally need somewhat longer intervals [x, x + ψ(x)] with a ψ(x) depending on the smoothness under consideration), even under the assumption of the Riemann Hypothesis.…”

Section: Corollarysupporting

confidence: 61%

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Multiplicative functions in short intervals

2016

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Abstract. We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of µ(n) in almost all intervals of the form [x, x + ψ(x)] with ψ(x) → ∞ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of x ǫ -smooth numbers in intervals of the form [x, x + c(ε) √ x], recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of λ(n)λ(n + 1), with λ(n) Liouville's function, is non-trivially bounded in absolute value by 1−δ for some δ > 0. This settles an old folklore conjecture and constitutes progress towards Chowla's conjecture. Fourth, we show that a (general) real-valued multiplicative function f has a positive proportion of sign changes if and only if f is negative on at least one integer and non-zero on a positive proportion of the integers. This improves on many previous works, and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.

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Section: Introductionsupporting

confidence: 76%

Section: Corollarysupporting

confidence: 61%

Multiplicative functions in short intervals

2016

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“…This is interesting especially in light of the recent result of Matomäki and Radziwiłł [15] (given there as a corollary of their much more general theorem on multiplicative functions) that for all > 0 there is a constant C = C( ) such that the intervals [x, x + C √ x] contain x -smooth numbers (this was previously known only for intervals [x, x + x 1/2 log 7/3+δ x], cf. [14]). The main idea in this paper is to combine their argument with the methods used for finding numbers with large prime factors in [x, x + x 1/2+ ], so that we can reduce the length of the interval as much as possible.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…x ǫ -smooth numbers (this was previously known only for intervals [x, x + x 1/2 log 7/3+δ x], cf. [14]). The main idea in the current manuscript is to combine their argument with the methods used for finding numbers with large prime factors in [x, x + x 1/2+ǫ ], so that we can reduce the length of the interval as much as possible.…”

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confidence: 99%

Large prime factors on short intervals

Merikoski

2019

Math. Proc. Camb. Phil. Soc.

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We show that for all large enough x the interval [x, x + x 1/2 log 1.39 x] contains numbers with a prime factor p > x 18/19 . Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x 1/2+ǫ ]. We also incorporate some ideas from Harman's book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman's sieve method.

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Another note on smooth numbers in short intervals

Abstract: Abstract. We prove that, for any positive constants δ and ε and every large enough x, the interval [x, x+ √ x(log x) 7/3+δ ] contains numbers whose all prime factors are smaller than x ε .

Cited by 2 publications

References 10 publications

Multiplicative functions in short intervals

Multiplicative functions in short intervals

Large prime factors on short intervals

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