The distribution of integers with a divisor in a given interval

Details The distribution of integers with a divisor in a given interval The distribution of integers with a divisor in a given interval

2004-01-18

arxiv.org/abs/math/0401223v5

Annals of Math. (2) 168 (2008), 367-433

Mathematics

Number Theory

Final version. Greatly simplified proof of Lemma 4.7 in Sec. 10, references updated, other minor corrections

Scientific paper

We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le \min(y^{C},x^{1/2-\epsilon}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics.

Affiliated with

Also associated with

No associations

Rating

The distribution of integers with a divisor in a given interval does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The distribution of integers with a divisor in a given interval, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The distribution of integers with a divisor in a given interval will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-28731