Letendre 2025 divisors integer short interval

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conjecture_1: The conjecture from the literature that Letendre records as Conjecture 1: for each fixed epsilon > 0 a constant bounds, for every n, the number of divisors of n in the closed window from n^{1/2} of length n^{1/2-epsilon}; the paper's Proposition 1 gives the cases 1/4 < epsilon < 1/2.

proposition_1: Letendre's unconditional bound: for 0 < theta < 1 and 0 < epsilon < theta^2, the number of divisors of n in [n^theta, n^theta + n^{theta^2-epsilon}] is << theta(1-theta)/epsilon + 1/(theta(1-theta)), a bound in which n does not appear.

theorem_1: Letendre's general bound for the number of divisors of n in [n^theta, n^theta + n^eta] with 0 < eta < theta < 1: at most tau(n) to the power 1 - xi(theta,eta), times V(n) log tau(n) / (theta(1-theta)), with an explicit five-case saving exponent xi equal to 1 when eta <= theta^2.

theorem_2: Letendre's lower bound for the constants of his Conjecture 2: if the conjecture holds for fixed 0 < theta < 1 and 0 < epsilon < theta, then k_epsilon(theta) >> sqrt(epsilon) (theta(1-theta))^{3/2} times (theta^theta (1-theta)^{1-theta})^{-1/epsilon}.

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Patrick Letendre, Divisors of an Integer in a Short Interval. arXiv preprint (2025). arXiv:2503.12146. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2503.12146), every other right reserved.

The paper studies D_n(X,Y), the number of divisors d of n with X <= d <= X+Y (p. 1). It records as Conjecture 1 (p. 1) a conjecture it attributes to the literature (Erdos and Rosenfeld; Chan): for fixed eps > 0 there is k_eps with D_n(n^{1/2}, n^{1/2-eps}) <= k_eps for each integer n >= 1. It proposes Conjecture 2 (p. 1): for fixed 0 < theta < 1 and 0 < eps < theta there is k_eps(theta) with D_n(n^theta, n^{theta-eps}) <= k_eps(theta) for each integer n >= 1; for eps < 1/2, Conjecture 1 is its case theta = 1/2. Theorem 1 (p. 2) is the general result: for fixed 0 < eta < theta < 1 and each integer n >= 2, D_n(n^theta, n^eta) << tau(n)^{1-xi(theta,eta)} V(n) log tau(n) / (theta(1-theta)), where V(n) is the largest exponent in the factorization of n and xi(theta,eta) is an explicit five-case exponent equal to 1 when eta <= theta^2. Theorem 2 (p. 2) shows that if Conjecture 2 holds, then k_eps(theta)

sqrt(eps) (theta(1-theta))^{3/2} (theta^{-theta}(1-theta)^{-(1-theta)})^{1/eps}, so the conjectured constants grow exponentially in 1/eps. Proposition 1 (p. 4) is the unconditional bounded case: for a fixed integer n >= 1, 0 < theta < 1 and 0 < eps < theta^2, D_n(n^theta, n^{theta^2-eps}) << theta(1-theta)/eps + 1/(theta(1-theta)), a bound in which n does not appear. Its proof rests on the lcm/gcd inequality of Lemma 1 (p. 2), which the paper takes from Cohen's Corollaire 1.4, applied to the divisors in the window. Section 6 (pp. 12-14) discusses a sieve approach to windows near sqrt(n) without proving Conjecture 1.

For #886, Proposition 1 at theta = 1/2 gives O(1/delta) divisors in [n^{1/2}, n^{1/2} + n^{1/4-delta}] for each fixed 0 < delta < 1/4, which answers the question for 1/4 < eps < 1/2, a range in which Erdos and Rosenfeld's bound already answers it; the full question is equivalent to the paper's Conjecture 1, which it leaves open. For #887, Proposition 1 reaches only windows of length n^{1/4-delta}, shorter than the question's C n^{1/4}, and settles no instance. The paper does not mention #873; a note posted in that problem's thread combines Proposition 1 with packing lemmas of its own in an argument that it says answers the question for every exponent above 1/4, and this card does not check that reduction.

Source: https://arxiv.org/abs/2503.12146.

Bears on. #886: Conjecture 1 is equivalent to a yes answer; Proposition 1 answers it for 1/4 < eps < 1/2. #887: Conjecture 1 would imply a yes answer; no result of the paper settles an instance. #873: Proposition 1 is an input to a posted note's argument for exponents above 1/4 (claim page); the paper does not treat the problem.

Results.

• Conjecture 1 (p. 1): for fixed eps > 0, D_n(n^{1/2}, n^{1/2-eps}) <= k_eps for each integer n >= 1; recorded from the literature and left open.
• Theorem 1 (p. 2): for fixed 0 < eta < theta < 1 and each n >= 2, D_n(n^theta, n^eta) << tau(n)^{1-xi(theta,eta)} V(n) log tau(n)/(theta(1-theta)), with the five-case exponent xi of (2.1), equal to 1 for eta <= theta^2.
• Theorem 2 (p. 2): for fixed 0 < theta < 1 and 0 < eps < theta, if Conjecture 2 holds then k_eps(theta) >> sqrt(eps) (theta(1-theta))^{3/2} (theta^{-theta}(1-theta)^{-(1-theta)})^{1/eps}.
• Proposition 1 (p. 4): for a fixed integer n >= 1, 0 < theta < 1 and 0 < eps < theta^2, D_n(n^theta, n^{theta^2-eps}) << theta(1-theta)/eps + 1/(theta(1-theta)).

Conjecture 2 is stated on the Theorem 2 page and Lemma 1 on the Proposition 1 page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.