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Relatively prime pairs in the Piatetski-Shapiro sequences

Yuta Suzuki

Abstract

In this paper, we prove an asymptotic formula for the number of relatively prime pairs in the Piatetski-Shapiro sequence of arbitrarily large order. This improves the result of Pimsert, Srichan and Tangsupphathawat (2023), the order of which was restricted to be $<\frac{3}{2}$. The key ingredients of the proof are a simple averaging trick and an extension of Deshouillers' result on the distribution of the Piatetski-Shapiro sequence in arithmetic progressions.

Relatively prime pairs in the Piatetski-Shapiro sequences

Abstract

In this paper, we prove an asymptotic formula for the number of relatively prime pairs in the Piatetski-Shapiro sequence of arbitrarily large order. This improves the result of Pimsert, Srichan and Tangsupphathawat (2023), the order of which was restricted to be . The key ingredients of the proof are a simple averaging trick and an extension of Deshouillers' result on the distribution of the Piatetski-Shapiro sequence in arithmetic progressions.
Paper Structure (5 sections, 10 theorems, 76 equations)

This paper contains 5 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary Lemmas
  4. The Piatetski-Shapiro sequences in arithmetic progressions
  5. Proof of the main theorem

Key Result

Theorem A

For $1\le c<\frac{3}{2}$ and $x\ge1$, we have where and the implicit constant depends only on $c$.

Theorems & Definitions (18)

  • Theorem A: Pimsert, Srichan and Tangsupphathawat PST
  • Theorem 1
  • Theorem B: Deshouillers Deshouillers:ComptesRendusDeshouillers:CubeFree
  • Theorem 2
  • Corollary 3
  • Remark 4
  • Theorem C
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 8 more