Analytic Number Theory pp 235-253 | Cite as

# On Gaps between Squarefree Numbers

## Abstract

A squarefree number is a positive integer not divisible by the square of an integer > 1. We investigate here the problem of finding small *h* = *h*(*x*) such that for *x* sufficiently large, there is a squarefree number in the interval (*x,x* + *h*]. This problem was originally investigated by Fogels [3]; he showed that for every ∈ > 0, *h* = *x* ^{2/5+∈} is admissible. Later Roth [9] reported elementary arguments of Davenport and Estermann showing respectively that one can take *h* ≫ *x* ^{1/3} and *h* ≫
*x* ^{1/3}(log *x*)^{-2/3} for sufficiently large choices of the implied constants. Roth then gave an elementary proof that *h* = *x* ^{1/4+ ∈} is admissible, and by applying a result of van der Corput, he showed that one can take *h*≫ *x* ^{3/13} (log *x*)^{4/13}Nair [6] later noted that the elementary proof could be modified to omit the ∈ in the exponent to get that *h*≫ *x* ^{1/4} is admissible, and more recently the first author [1] showed that one could obtain the result
*h*≫*x* ^{3/13} by elementary means. Using further exponential sum techniques, Richert [8], Rankin [7], Schmidt [10], and Graham and Kolesnik [4] obtained the improvements *h* ≫ *x* ^{2/9}log *x*, *h* = *x* ^{ θ } ^{+€} where *θ* = 0.221982…, *θ* = 109556/494419 = 0.221585…, and *θ* = 1057/4785 = 0.2208986…, respectively. The authors investigated the problem further.

## Keywords

Elementary Proof Divided Difference Implied Constant Prime Number Theorem Elementary Argument## Preview

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