1 results
NOTE ON A DYNAMICAL GENERALISATION OF THE PRIME NUMBER THEOREM FOR ARITHMETIC PROGRESSIONS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 27 February 2025, pp. 1-8
-
- Article
- Export citation
-
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.
