Ford 2008 distribution integers divisor given interval

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corollary_2: For c > 1 and 1/(c-1) <= y <= x/c, H(x,y,cy) is of order x/((log Y)^delta (log log Y)^(3/2)) with Y = min(y, x/y) + 3, and the density epsilon(y,cy) is of order 1/((log y)^delta (log log y)^(3/2)), the constants depending on c.

corollary_3: A(x), the number of n up to x that can be written as n = m_1 m_2 with each m_i at most sqrt(x), is of order x/((log x)^delta (log log x)^(3/2)), the order of the number of distinct entries of the multiplication table.

corollary_5: For x >= 3, the average over n up to x of tau^+(n), the number of k with a divisor of n in (2^k, 2^(k+1)], is of order (log x)^(1-delta)/(log log x)^(3/2).

corollary_7: For every lambda > 1 and r >= 1 the density of integers with exactly r divisors in (y, lambda y] is bounded below by a constant times the density of those with at least one, while for z/y tending to infinity the ratio tends to 0; this refutes Erdos's Conjecture 1 as the paper states it.

theorem_1: The order of magnitude of H(x,y,z), the number of n up to x with a divisor in (y,z], for all 1 <= y <= z <= x, in four trivial ranges and two main ones: H/x is of order eta, beta/(max(1,-xi)(log y)^G(beta)), u^delta (log 2/u)^(-3/2) or 1 as z grows, when y <= sqrt(x).

theorem_2: For y_0 <= y <= sqrt(x), z >= y + 1 and x/log^10 z <= Delta <= x, the number of n in (x - Delta, x] with a divisor in (y,z] is of order (Delta/x) H(x,y,z); the short-interval form of Theorem 1 used to prove its part (vi).

theorem_3: For y_0 <= y <= sqrt(x), y + 1 <= z <= x and x/log y <= Delta <= x, the number of squarefree n in (x - Delta, x] with a divisor in (y,z] is of order (Delta/x) H(x,y,z) when z >= y + K y^(1/5) log y, and also, with constants depending on g, when y + (log y)^(2/3) <= z <= y + K y^(1/5) log y and (y,z] holds at least g(z - y) squarefree numbers.

theorem_4: For c > 0, y >= y_0(c), y + 1 <= z <= x^(5/8) and yz <= x^(1-c), the proportion of the integers with a divisor in (y,z] that have exactly one such divisor is of order log log(z/y + 10)/log(z/y + 10), the constants depending on c.

theorem_5: For r >= 2, c > 0, y >= y_0(r,c), z <= x^(5/8) and yz <= x^(1-c), the proportion H_r/H of integers with exactly r divisors in (y,z] is at least a constant times max(1,-xi)/sqrt(log log y) and at most 1 for z_0(y) <= z <= 10y, has order (log log(z/y))^(nu(r)+1)/log(z/y) for 10y <= z <= y^C, and is at least a constant times (log log y)^(nu(r)+1)/log z for y^2 <= z <= x^(5/8).

theorem_6: For a fixed non-zero integer lambda, 1 <= y <= sqrt(x) and y + 1 <= z <= x, the number of q + lambda up to x, q prime, with a divisor in (y,z] is at most a constant times H(x,y,z)/log x when z >= y + (log y)^(2/3), and times (x/log x) times the sum of 1/phi(d) over y < d <= z otherwise.

theorem_7: For fixed lambda, a, b with lambda non-zero and 0 <= a < b <= 1, the number of q + lambda up to x, q prime, with a divisor in (x^a, x^b] is at least a constant times x/log x, the constant depending on a, b and lambda.

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Kevin Ford, The distribution of integers with a divisor in a given interval. Annals of Mathematics (2) 168 (2008), 367-433. arXiv:math/0401223, doi:10.4007/annals.2008.168.367. The arXiv record names arXiv's non-exclusive distribution license (arXiv:math/0401223), every other right reserved.

Theorem 1 determines the order of magnitude of H(x,y,z), the number of n <= x with a divisor in (y,z], for all x, y, z, with Corollary 1 showing the normalized count depends only on the sizes of log(z/y), log y and log(x/z). Corollary 2 specializes to short intervals: for c > 1 and 1/(c-1) <= y <= x/c, H(x,y,cy) is of order x/((log Y)^delta (log log Y)^{3/2}) with Y = min(y, x/y) + 3, and the density epsilon(y,cy) is of order 1/((log y)^delta (log log y)^{3/2}), the constants depending on c, where delta = 1 - (1 + log log 2)/log 2 = 0.086071...; this sharpens Erdos's 1960 estimate epsilon(y,2y) = (log y)^{-delta+o(1)}. Corollary 3 gives the same order x/((log x)^delta (log log x)^{3/2}) for A(x), the number of n <= x of the form m_1 m_2 with each m_i <= sqrt(x). Theorem 2 shows H(x,y,z) - H(x-Delta,y,z) is of order (Delta/x) H(x,y,z) for y_0 <= y <= sqrt(x), z >= y + 1 and Delta down to x/log^{10} z, i.e. the same count holds in long subintervals, and Theorem 3 gives a squarefree analog. Theorems 4 and 5 bound H_r(x,y,z), the count with exactly r divisors in (y,z], relative to H(x,y,z): Theorem 4 gives H_1/H of order log log(z/y+10)/log(z/y+10) when c > 0, y >= y_0(c), y + 1 <= z <= x^{5/8} and yz <= x^{1-c}, and Theorem 5 bounds H_r/H for r >= 2, giving its order when 10y <= z <= y^C, and lower bounds for z_0(y) <= z <= 10y and y^2 <= z <= x^{5/8}. Corollary 7 deduces that epsilon_r(y,lambda y) >>{r,lambda} epsilon(y,lambda y) for every lambda > 1 and r >= 1, while epsilon_r(y,z)/epsilon(y,z) -> 0 as z/y -> infinity; so Erdos's Conjecture 1 is false, Tenenbaum's Conjecture 3 is true, and his Conjecture 2 is true provided z >= y + y/(log y)^{log 4 - 1 - b} for a fixed b > 0. The method combines uniform order statistics with sieve-style reductions to volume and integral estimates. For #859 this is the material upper-route source: it pins the order of H(x,y,z) and of epsilon(y,cy), and with y = t/(log t)^2 and z = t, the case 2y <= z <= y^2 of Theorem 1 (v) gives the integers with a divisor in (t/(log t)^2, t) density of order (log t)^{-delta} (log log t)^{delta - 3/2}. Since Erdos's 1970 split caps the rest of A_t, the n with no such divisor, at density 2/log t through (32), this bounds d_t from above by that order only; the paper never estimates d_t itself and gives no lower bound, so it leaves the conjectured clean power-of-log asymptotic open. For #450 Corollary 2 supplies the order of the density of integers with a divisor in (n,2n], the fraction the problem's every-x reading is compared with; Theorem 2 gives the matching count only in intervals (x - Delta, x] with Delta >= x/log^{10} z, which grow with x, so it does not bound the window length the problem asks about. For #446, Corollary 2 at c = 2 gives the order of the density of the integers with a divisor in (n,2n], and Corollary 7 with r = 1 and lambda = 2 refutes Erdos's expectation delta_1(n) = o(delta(n)), which the paper states as Conjecture 1. For #896, Corollary 3 bounds the number of distinct entries of the N x N multiplication table, and so the maximum of F(A,B), by N^2/((log N)^delta (log log N)^{3/2}) up to a constant. Theorem 4, with Corollary 2, is the estimate for integers with exactly one divisor in (y,2y] that the accepted lower-bound construction for #896 cites, and the input Cambie's Claim 4 cites for the local maxima of delta_1(n,m) in #692. Corollary 5, (1/x) sum{n<=x} tau^+(n) of order (log x)^{1-delta}/(log log x)^{3/2} for x >= 3, answers the companion estimate that #448 mentions, not the question #448 states.

Source: https://arxiv.org/abs/math/0401223.

Bears on. #446: Corollary 2 gives the order of delta(n), and Corollary 7 with r = 1 and lambda = 2 refutes delta_1(n) = o(delta(n)). #448: Corollary 5 gives the order of the companion sum of tau^+(n), not the question the problem states. #450: Corollary 2 gives the order of the density the problem's every-x reading is compared with. #692: Theorem 4 is an input that Cambie's Claim 4 cites. #859: Theorem 1 (v) gives the order of the density of integers with a divisor in (t/(log t)^2, t]; the paper never estimates d_t. #896: Corollary 3 gives the upper bound for the maximum of F(A,B); Corollary 2 and Theorem 4 are the estimates the lower-bound construction cites.

Results. Labels and pages are those of the Annals print.

• Theorem 1 (p. 371): the order of H(x,y,z) for all 1 <= y <= z <= x.
• Corollary 2 (p. 372): H(x,y,cy) and epsilon(y,cy).
• Theorem 2 (p. 372): H(x,y,z) in intervals (x - Delta, x] with Delta >= x/log^{10} z.
• Theorem 3 (p. 372): the squarefree count H*(x,y,z) in such intervals.
• Corollary 3 (p. 373): the multiplication-table count A(x).
• Corollary 5 (p. 373): the mean of tau^+(n).
• Theorem 4 (p. 375): the order of H_1(x,y,z)/H(x,y,z).
• Theorem 5 (p. 376): bounds for H_r(x,y,z)/H(x,y,z), r >= 2.
• Corollary 7 (p. 376): epsilon_r against epsilon, and the conjectures of Erdos and Tenenbaum.
• Theorem 6 (p. 378): upper bounds for shifted primes.
• Theorem 7 (p. 378): a lower bound for shifted primes.

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