ON THE LOCAL BEHAVIOR OF 9(x,y)

ty(x,y) denotes the number of positive integers < i and free of prime factors > y. In the range y > exp((loglogx)5T3+£), *9(x, y) can be well approximated by a "smooth" function, but for y < (logx)2-e, this is no longer the case, since then the influence of irregularities in the distribution of primes becomes apparent. We show that *¡>(x,y) behaves "locally" more regular by giving a sharp estimate for ~i/(cx,y)/^/(x,y), valid in the range x > y > 4 log x, 1 < c < y. Introduction. The function W(x, y) denotes the number of positive integers < x having no prime factors > y. Many arithmetic problems require accurate estimates for ty(x, y), and the study of this function has been the object of numerous articles. A survey of the most important results as well as an extensive bibliography can be found in Norton's memoir [8]. It turns out that *(x, y)/x, i.e. the "probability" for a positive integer < x to be free of prime factors > y, depends essentially on the ratio log x/ log y. Dickman [4] showed, that for every fixed u > 0, the limit limv_00 ^(yu,y)/yu exists and equals p(u), where p (the "Dickman function") is defined by p(u) = 1 (0 < u < 1), p continuous at 1, .. . p(u 1) , ,. //(«) = u (u>l). De Bruijn [3] established the quantitative estimate (i) nyu,y) = yuP(u){i + oe(^^ for the range y > 2, 1 < u < (logy)3/5_£, where e is any fixed positive number. In [7], this range was extended to y > 2, 1 < u < exp(logy)3/5_£. It is natural to ask whether these results can be further improved and, in particular, for what range relation (1) can possibly hold. The upper bound u < exp(logy)3//5_£ in the last mentioned result arises from the sharpest known form of the prime number theorem and could be replaced by u < y1//2_£, if the Riemann hypothesis is assumed. However, one cannot go much further, and it seems likely that for u sb ^/y relation (1) no longer holds. In fact, it appears that for u > y1/2+e, ty(yu,y) becomes strongly dependent on the irregularities in the distribution of primes and cannot be well approximated by "smooth" functions. In Theorem 3 we shall give an estimate, which clearly exhibits the connection between ^{yuiy) and the distribution properties of the prime numbers. Received by the editors October 7, 1985. 1980 Mathematics Subject Classification. Primary 10H15.