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On moments of gaps between consecutive squarefree numbers

Tsz Ho Chan

Abstract

Let $s_1, s_2, s_3, \cdots$ be the set of squarefree numbers in ascending order. In this paper, we prove that the following asymptotic on moments of gaps between squarefree numbers \[ \sum_{s_{k+1} \le x} (s_{k+1} - s_k)^γ\sim B(γ) x \; \; \mbox{ with some constant} \; \; B(γ) > 0 \] is true for $0 \le γ< 3.75$. This improves the previous best range $0 \le γ< 3.6875$.

On moments of gaps between consecutive squarefree numbers

Abstract

Let be the set of squarefree numbers in ascending order. In this paper, we prove that the following asymptotic on moments of gaps between squarefree numbers is true for . This improves the previous best range .
Paper Structure (4 sections, 5 theorems, 59 equations)

This paper contains 4 sections, 5 theorems, 59 equations.

Table of Contents

  1. Introduction and Main Result
  2. A summary of Huxley's work
  3. Improvement on Case 1(a)
  4. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1

For $0 \le \gamma < 3.75$, with $B(\gamma)$ defined by (Bgamma).

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • proof
  • Proposition 2
  • proof : Proof of Theorem \ref{['mainthm']}