On the difference between consecutive squarefree integers. (English) Zbl 0656.10041

Graham, S. W.; Kolesnik, G.

Acta Arith. 49, No. 5, 435-447 (1988).

The authors estimate the difference between consequtive squarefree numbers. Denoting by \(S_ n\) the nth squarefree number they prove that \[ (*)\quad S_{n+1}-S_ n=O(n^{a+\epsilon}),\quad a=1057/4785 \] improving earlier results by Fogels \((a=2/5)\), Roth \((a=3/13)\), Richert \((a=2/9)\), Rankin \((a=0.2219821...)\) and P. G. Schmidt \((a=0.2215834...)\). In the first part of the proof they show how results of type (*) follow from a sufficiently sharp estimate of exponential sums of the form \[ S=\sum_{K\leq k\leq K_ 1\leq 2K}\sum_{N\leq n\leq N_ 1\leq 2N}e(kyn^{-b}). \] Next, they establish such an estimate (theorem 1): For \(b=2\) and \(y=x\) or \(b=\) and \(y=x^{1/2}\) \[ | S| \ll K x^{1057/4785} \] provided \(x^{1057/4785}\leq N\leq x^{1/3}\) and \(K\leq N^{1+\epsilon} x^{-1057/4785}\).

Reviewer: J.Kaczorowski

MSC:

11N05	Distribution of primes
11L40	Estimates on character sums

Keywords:

difference between consequtive squarefree numbers; sharp estimate of exponential sums