The distribution of special subsets of positive integers is a central topic of research in analytic number theory. In this paper we present a survey of recent progress on the distribution of B-free numbers in short intervals and some of its applications. | Turk J Math 30 (2006) , 293 – 308. ¨ ITAK ˙ c TUB A Survey on the Distribution of B-free Numbers Emre Alkan and Alexandru Zaharescu Abstract In this paper we present a survey of recent progress on the distribution of B-free numbers in short intervals and some of its applications. Key Words: B-free numbers, linear forms. 1. Introduction The distribution of special subsets of positive integers is a central topic of research in analytic number theory. In particular, extensive research has been done on the distribution of prime numbers. For an account on recent developments the reader may consult the nice survey of Yıldırım [25]. For the purpose of some applications one is motivated to consider the distribution of certain sequences of numbers defined by milder divisibility constraints. The notion of a B-free number, introduced by Erd¨os in [8], generalizes that of a square-free number. Given a sequence B of positive integers 1 0. N→∞ N bk () k=1 By taking B to be the sequence of squares of all the prime numbers, the set of B-free numbers coincides with the set of square-free numbers. Erd¨ os [8] proved that for some c c 0 there exists NB, such that for any N ≥ NB, the interval os’ claim for [N, N + N ] contains at least one B-free number. Szemer´edi [23] proved Erd¨ 1 9 all c > 2 . This was improved to c > 20 by Bantle and Grupp [6], using the work of Iwaniec and Laborde [13]. A further improvement is due to Wu [24], who proved that #{N ≤ n ≤ N + N c : n is B-free} B,c N c () for all c > 17 41 , using the work of Fouvry and Iwaniec [10] on exponential sums with monomials. Zhai [26] obtained the same result for c > 33 80 . The best result to date . Assuming is due to Sargos and Wu [20], who reduced the value of c to c > 40 97 the ABC-Conjecture Granville [11] proved Erd¨ os’ Conjecture in the case of square-free numbers. Unconditionally, Filaseta and Trifonov [9] proved that for N large enough and for all c > 15 , the interval [N, N + N c ] .
