New small gaps between squarefree numbers

2021/10/19 by Tsz Ho Chan, Chan, Tsz Ho
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2110.09990

openalex publication_date 2021/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we show that, for some constant C > 0, the interval (x, x + C x^5/26] always contains a squarefree number when x is sufficiently large (in terms of C). Our improvement comes from establishing asymptotic relations between the shifts a and b when m n² ≈ (m - a) (n + b)² We apply them to study quadruples (m + a₁) (n - b₁)² ≈ m n² ≈ (m - a₂)(n + b₂)² ≈ (m - a₂ - a₃)(n + b₂ + b₃)² and generalize Roth differencing and Filaseta-Trifonov differencing by allowing b₁ to be different from b₃. We also introduce a new differencing and exploit the interplay among these three differencings.

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• Small gaps in coefficients of L-functions and B-free numbers in small intervals