Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence

Jean-Marc Deshouillers (), Michael Drmota and Clemens Müllner
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Jean-Marc Deshouillers: Université de Bordeaux, Bordeaux INP, CNRS, Institut de Mathématiques de Bordeaux, UMR 5251
Michael Drmota: TU Wien, Institute of Discrete Mathematics and Geometry
Clemens Müllner: TU Wien, Institute of Discrete Mathematics and Geometry

A chapter in Number Theory in Memory of Eduard Wirsing, 2023, pp 91-98 from Springer

Abstract: Abstract We show that in any Piatetski-Shapiro sequence ⌊ n c ⌋ n $$\left (\lfloor n^c \rfloor \right )_n$$ with c in ( 1 , + ∞ ) ∖ ℕ $$(1, +\infty )\backslash \mathbb {N}$$ , there exist long subsequences of consecutive elements no pair of which are coprime, whereas for any c in ( 1 , 2 ) $$(1, 2)$$ , there exist infinitely many n such that all the elements in { ⌊ n c ⌋ , ⌊ ( n + 1 ) c ⌋ , … , ⌊ ( n + H ) c ⌋ } $$\{\lfloor n^c \rfloor , \lfloor (n+1)^c \rfloor , \ldots , \lfloor (n+H)^c \rfloor \}$$ are pairwise coprime for H almost as large as min ( c − 1 , 1 − c ∕ 2 ) log n $$\min (c-1,1-c/2)\log n$$ .

Keywords: Piatetski-Shapiro sequences; Coprimality; Congruences; Distribution modulo 1; Trigonometric sums (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-31617-3_7

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DOI: 10.1007/978-3-031-31617-3_7

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