Squarefree numbers in short intervals

2024/01/25 by Mayank Pandey, Pandey, Mayank
Mathematics · #11N25 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.13981

openalex publication_date 2024/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that there exists η> 0 such that the interval [X, X + X^{\frac 15 - η}] contains a squarefree number for all large X. This improves on an earlier result of Filaseta and Trifonov who showed that there is a squarefree number in [X, X + cX^{\frac 15}log X] for some c > 0 and all large X. We introduce a new technique to count lattice points near curves, which we use to bound in critical ranges the number of integers in a short interval divisible by a large square. This uses as an input Green and Tao's quantitative version of Leibman's theorem on the equidistribution of polynomial orbits in nilmanifolds.

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