The distribution of r-tuples of square-free numbers

Kai-Man Tsang

doi:10.1112/S0025579300011049

Abstract

We use the Buchstab-Rosser sieve to derive an asymptotic formula for the distribution of those integers n for which the r numbers n + l1, n + l2,…, n + lr are all square-free. Our error estimate sharpens a similar result of Hall and is uniform in both r and maxl 1≤i≤r|li|.

References

1.Hall, R. R.. Square-free numbers on short intervals. Mathematika, 29 (1982), 7–17.CrossRef Google Scholar

2.Heath-Brown, D. R.. Square sieve and consecutive square-free numbers. Math. Ann., 266 (1984), 251–259.CrossRef Google Scholar

3.Hooley, C.. Applications of Sieve Methods to the Theory of Numbers (Cambridge, 1976).Google Scholar

4.Ramanujan, S.. The normal number of prime factors of a number n. Quarterly Jour, of Math., 48 (1917), 76–92.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

TOLEV, D. I. 2006. ON THE DISTRIBUTION OF r-TUPLES OF SQUAREFREE NUMBERS IN SHORT INTERVALS. International Journal of Number Theory, Vol. 02, Issue. 02, p. 225.

Dartyge, Cécile and Sárközy, András 2007. On Pseudo-Random Subsets of the Set of the Integers Not Exceeding N. Periodica Mathematica Hungarica, Vol. 54, Issue. 2, p. 183.

Jancevskis, Martin 2009. Convergent sieve sequences in arithmetic progressions. Journal of Number Theory, Vol. 129, Issue. 6, p. 1595.

Pleasants, Peter A. B. and Huck, Christian 2013. Entropy and Diffraction of the $$k$$ k -Free Points in $$n$$ n -Dimensional Lattices. Discrete & Computational Geometry, Vol. 50, Issue. 1, p. 39.

Zhang, Ting and Liu, Huaning 2014. On the Distribution of r-Tuples of k-Free Numbers. Journal of Numbers, Vol. 2014, Issue. , p. 1.

Peckner, Ryan 2015. Uniqueness of the measure of maximal entropy for the squarefree flow. Israel Journal of Mathematics, Vol. 210, Issue. 1, p. 335.

Avdeeva, M Cellarosi, Francesco and Sinai, Yakov Grigor'evich 2016. Ergodic and statistical properties of $\mathscr{B}$-free numbers. Теория вероятностей и ее применения, Vol. 61, Issue. 4, p. 805.

Avdeeva, M. Cellarosi, F. and Sinai, Ya. G. 2017. Ergodic and Statistical Properties of $\B$-Free Numbers. Theory of Probability & Its Applications, Vol. 61, Issue. 4, p. 569.

Vatwani, Akshaa 2019. Variants of equidistribution in arithmetic progression and the twin prime conjecture. Mathematische Zeitschrift, Vol. 293, Issue. 1-2, p. 285.

Gorodetsky, Ofir Matomäki, Kaisa Radziwiłł, Maksym and Rodgers, Brad 2021. On the variance of squarefree integers in short intervals and arithmetic progressions. Geometric and Functional Analysis, Vol. 31, Issue. 1, p. 111.

Sanna, Carlo 2022. On the exponents in the factorizations ofrconsecutive numbers. Quaestiones Mathematicae, Vol. 45, Issue. 8, p. 1221.

M. Nunes, Ramon 2022. Moments of the distribution of k$k$‐free numbers in short intervals and arithmetic progressions. Bulletin of the London Mathematical Society, Vol. 54, Issue. 4, p. 1282.

Qin, Huan and Ye, Yangbo 2024. Algorithms of the Möbius function by random forests and neural networks. Journal of Big Data, Vol. 11, Issue. 1,

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The distribution of r-tuples of square-free numbers

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This article has been cited by the following publications. This list is generated based on data provided by Crossref.

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The distribution of r-tuples of square-free numbers

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