The distribution of integers with a divisor in a given interval

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 H r (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 H 1 (x, y, z) for all x, y, z satisfying z ≤ x 1/2−ε . For every r ≥ 2, C > 1 and ε > 0, we determine the order of magnitude of H r (x, y, z) uniformly for y large and y + y/(log y) log 4−1−ε ≤ z ≤ min(y C , x 1/2−ε ). As a consequence of these bounds, we settle a 1960 conjecture of Erdős and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.

Permutations Fixing ak-set

“…This last problem has a long history, originating as a problem of Besicovitch [Bes34] in 1934, and was solved (up to a constant factor) by the second author [For08a,For08b]. In those papers it was shown thatĩ(n, k) ≍ k −δ (1 + log k) −3/2 uniformly for k n/2, where δ is the constant mentioned above.…”

“…Instead of directly estimating the probability of a single number lying in L (X k ), however, we apply a local-to-global principle used in [For08a,For08b] to reduce the problem to studying the size of L (X k ). We expect a positive proportion of the elements of L (X k ) to lie in the range [ 1 10 k, 10k] (say).…”

The multiplication table problem for bipartite graphs

“…and then by Tenenbaum [22]. Despite its innocuous appearance, the multiplication table problem has been settled only recently; a deep result of Ford [14] asserts that as n → ∞,…”

Boneh-Boyen Signatures and the Strong Diffie-Hellman Problem

“…Out of these 900 curves, all curves except one (the curve with 1461501641662054988059088728056207736278975404329 points) admit a divisor for which the runtime predicted by Equation (1) is within a factor of 4 of the optimal time. In addition, Ford [14] has shown asymptotically that a large proportion of primes p admit a divisor d within the interval required for our algorithm. These results indicate that pairing-friendly curves are unlikely to resist the SDH algorithm unless specifically chosen with this property in mind.…”