tailieunhanh - Đề tài " The distribution of integers with a divisor in a given interval "

We determine the order of magnitude of H(x, y, z), the number of integers n ≤ x having a divisor in (y, z], for all x, y and z. We also study Hr (x, y, z), the number of integers n ≤ x having exactly r divisors in (y, z]. When r = 1 we establish the order of magnitude of H1 (x, y, z) for all x, y, z satisfying z ≤ x1/2−ε . For every r ≥ 2, C 1 and ε 0, we determine the order of magnitude of Hr (x, y, z) uniformly. | Annals of Mathematics The distribution of integers with a divisor in a given interval By Kevin Ford Annals of Mathematics 168 2008 367 433 The distribution of integers with a divisor in a given interval By Kevin Ford Abstract We determine the order of magnitude of H x y z the number of integers n x having a divisor in y z for all x y and z. We also study Hr x y z the number of integers n x having exactly r divisors in y z . When r 1 we establish the order of magnitude of H1 x y z for all x y z satisfying z x1 2- . For every r 2 C 1 and 0 we determine the order of magnitude of Hr x y z uniformly for y large and y y logy log4-1_e z min yC x1 2- . As a consequence of these bounds we settle a 1960 conjecture of Erd os and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics. Contents 1. Introduction 2. Preliminary lemmas 3. Upper bounds outline 4. Lower bounds outline 5. Proof of Theorems 1 2 3 4 and 5 6. Initial sums over L a ơ and Ls a ơ 7. Upper bounds in terms of S t ơ 8. Upper bounds reduction to an integral 9. Lower bounds isolated divisors 10. Lower bounds reduction to a volume 11. Uniform order statistics 12. The lower bound volume 13. The upper bound integral 14. Divisors of shifted primes References 1. Introduction For 0 y z let T n y z be the number of divisors d of n which satisfy y d z. Our focus in this paper is to estimate H x y z the number of positive integers n x with T n y z 0 and Hr x y z the number of 368 KEVIN FORD n x with T n y z r. By inclusion-exclusion H w x -1 k iEk z he-4 but this is not useful for estimating H x y z unless z y is small. With y and z fixed however this formula implies that the set of positive integers having at least one divisor in y z has an asymptotic density . the limit . H x y z y z lim v y XU x exists. Similarly the exact formula Hr x y z X r X Mdi . k r y d1 --- dfc z L J implies the existence of . Hr x y z r y z lim rv 7 XU x for every