The distribution of integers with a divisor in a given interval

By Kevin Ford. 2004

234 citations recorded by the source corpus.

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]. Whenr = 1 we establish the order of magnitude ofH1(x;y;z) for allx;y;z satisfying z x 1=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) log 4 1 " z min(y C ;x 1=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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/cb30855c2659c333b63d72a501ac7d686042310c

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Modular hyperbolas (2011, methodology): “Indeed, it has been noticed in [91] that well-known estimates for integers with a divisor in a given interval (see [73, 106]) immediately imply that for any ε > 0 almost all residue classes modulo m are not of the form xy (mod m) with 1 6 x, y 6 m(logm) where κ is given by (9).”
On the energy variant of the sum-product conjecture (2016, background): “(See [11] proving the explicit value of s = 0.”
Sieving by large integers and covering systems of congruences (2005, background): “In particular, at most finitely many integers H ∈ H are covering.”
Refined estimates concerning sumsets contained in the roots of unity (2019, background): “Assuming (as we may) that M ď N, then M is a divisor d of p´1 with 1 100 ? x ď d ď ? x provided p is suﬃciently large. So with z “ ? x and y “ ? x{100 we have Ppx,y,zq ! Hpx,y,zq logx by Theorem 6 of [F]. Next by Theorem 1 part (v) of [F], with u “ logp100q{logpyq, we have Hpx,y,zq “ opxq (in a stro”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, background): “,√p/3} ⊂ Z then T (A) ≫ |A|4 log |A|, for some absolute 0 < γ < 1, due to Ford [11].”