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NOTE ON A DYNAMICAL GENERALISATION OF THE PRIME NUMBER THEOREM FOR ARITHMETIC PROGRESSIONS

Part of: Multiplicative number theory

Published online by Cambridge University Press:  27 February 2025

BIAO WANG Open the ORCID record for BIAO WANG [Opens in a new window]
BIAO WANG*
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650500, PR China
*
e-mail: [email protected]
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Abstract

Bergelson and Richter [‘Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions’, Duke Math. J. 171(15) (2022), 3133–3200] established a new dynamical generalisation of the prime number theorem (PNT) and the PNT for arithmetic progressions. Let $h\ge 1, k\ge 2$. Mirsky [‘Note on an asymptotic formula connected with r-free integers’, Quart. J. Math. Oxford Ser. 18 (1947), 178–182] showed that the numbers n such that $n+l_1,\ldots , n+l_h$ are k-free have a natural density for any given nonnegative integers $l_1,\ldots , l_h$. In this note, we show that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.

Keywords

prime number theoremnumber of prime factorsk-free integersunique ergodicity

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 8
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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References

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