On Gaps between Squarefree Numbers
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 ; he showed that for every ∈ > 0, h = x 2/5+∈ is admissible. Later Roth  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)4/13Nair  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  showed that one could obtain the result h≫x 3/13 by elementary means. Using further exponential sum techniques, Richert , Rankin , Schmidt , and Graham and Kolesnik  obtained the improvements h ≫ x 2/9log 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.
KeywordsElementary Proof Divided Difference Implied Constant Prime Number Theorem Elementary Argument
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