The divisor function at consecutive integers

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lemma_1: Heath-Brown's Key Lemma, cited by Hildebrand: for every positive integer k there are positive integers a_1 < ... < a_k such that each difference a_j - a_i divides gcd(a_i, a_j) and the divisor counts satisfy d(a_j) d(a_i/(a_j - a_i)) = d(a_i) d(a_j/(a_j - a_i)).

theorem_1: Hildebrand's theorem that for all sufficiently large x the number of integers n <= x with d(n) = d(n+1) is at least a constant times x(log log x)^{-3}.

theorem_2: Hildebrand's theorem that for positive integers d_1, ..., d_7 and all sufficiently large x, the number of n <= x with d(n+1)/d(n) = d_j/d_i, summed over the pairs 1 <= i < j <= 7, is at least a constant times x(log log x)^{-3}.

theorem_3: Hildebrand's theorem that the set E of limit points of log(d(n+1)/d(n)) has positive lower Lebesgue density in [0, x] and in [-x, 0], and contains an interval [-δ, δ] for some δ > 0.

theorem_4: Hildebrand's theorem that for nonnegative integers d_1, ..., d_7 and all sufficiently large x, the number of n <= x with Omega(n+1) - Omega(n) = d_j - d_i, summed over the pairs 1 <= i < j <= 7, is at least a constant times x(log log x)^{-3}.

theorem_5: Hildebrand's theorem that the set A of integers a such that Omega(n) - Omega(n+1) = a for infinitely many n has positive lower density.

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Hildebrand, Adolf, The divisor function at consecutive integers. Pacific J. Math. 129 (1987), no. 2, 307--319, doi:10.2140/pjm.1987.129.307. The copy read prints "Copyright © 1987 by Pacific Journal of Mathematics" on the journal's editorial page appended to the download (PDF p. 16 of 17), every other right reserved.

Theorem 1 (p. 307) shows that for all sufficiently large $x$ the number of $n\le x$ with $d(n)=d(n+1)$ is $\gg x(\log\log x)^{-3}$. This improves Heath-Brown's $\gg x(\log x)^{-7}$, display (1.1), and falls a power of $\log\log x$ short of the order $x(\log\log x)^{-1/2}$ that the paper calls the conjectured right one, matching the upper bound (1.3) of Erdős, Pomerance and Sárközy (p. 307). The proof combines Heath-Brown's Key Lemma (Lemma 1) with an idea of Erdős, Pomerance and Sárközy and the sieve estimate Lemma 2, the sharpest known of its type; the paper notes that Lemma 2 with $g=r=2$, as has been conjectured, would give $\gg x(\log\log x)^{-1/2}$ (p. 308). Theorem 2 (p. 308) proves the more general bound: for positive integers $d_1,\ldots,d_7$, the counts of $n\le x$ with $d(n+1)/d(n)=d_j/d_i$, summed over $1\le i<j\le7$, are $\gg x(\log\log x)^{-3}$; with all $d_i$ equal this is Theorem 1. From it Theorem 3 (p. 308) deduces that the set $E$ of limit points of $\log(d(n+1)/d(n))$ has positive lower density in $[0,x]$ and in $[-x,0]$ (the proof gives measure at least $x/36$ in each, p. 319) and contains an interval $[-\delta,\delta]$ for some $\delta>0$. The paper presents this as partly settling Erdős's conjecture that every positive real is a limit point of $d(n+1)/d(n)$, and says that before it only $0$ was known to lie in $E$. Theorems 4 and 5 (p. 309) are the analogues for $\Omega(n)$, stated without separate proof: Theorem 4 is the bound of Theorem 2 for $\Omega(n+1)-\Omega(n)=d_j-d_i$ with nonnegative $d_i$, and Theorem 5 says the set of integers $a$ with $\Omega(n)-\Omega(n+1)=a$ for infinitely many $n$ has positive lower density.

Source: https://msp.org/pjm/1987/129-2/p06.xhtml.

Read status: claims checked for Theorems 1 to 5 and Lemma 1, read clause by clause on the page images of the print; the proofs of Theorem 2 (§4, with (4.10) only sketched in the paper) and Theorem 3 (§5) followed. Lemma 1 is quoted from Heath-Brown and Lemma 2 adapted from Halberstam and Richert; their proofs were not read. Theorems 4 and 5 have no proof in the paper. Nothing here is independently reviewed.

Bears on. #946: Theorem 1 (p. 307) gives, for all sufficiently large $x$, at least $c\,x(\log\log x)^{-3}$ integers $n\le x$ with $\tau(n)=\tau(n+1)$, which answers the problem's question yes; the problem's claim page for this paper records it. #964: Theorem 3 (p. 308) shows that the limit points of $\tau(n+1)/\tau(n)$ include an interval $[e^{-\delta},e^{\delta}]$ and that their logarithms have positive lower density in $[0,x]$ and in $[-x,0]$; it does not decide whether the ratios are dense in $(0,\infty)$.

Results.

• Theorem 1 (p. 307): $#{n\le x:d(n)=d(n+1)}\gg x(\log\log x)^{-3}$ for sufficiently large $x$.
• Theorem 2 (p. 308): for positive integers $d_1,\ldots,d_7$, the counts of $n\le x$ with $d(n+1)/d(n)=d_j/d_i$, summed over $1\le i<j\le7$, are $\gg x(\log\log x)^{-3}$.
• Theorem 3 (p. 308): the limit points of $\log(d(n+1)/d(n))$ have positive lower density in $[0,x]$ and in $[-x,0]$ and contain some $[-\delta,\delta]$.
• Theorem 4 (p. 309): the analogue of Theorem 2 for $\Omega(n+1)-\Omega(n)=d_j-d_i$ with nonnegative integers $d_i$.
• Theorem 5 (p. 309): the integers $a$ with $\Omega(n)-\Omega(n+1)=a$ for infinitely many $n$ have positive lower density.
• Lemma 1 (p. 309): Heath-Brown's Key Lemma, taken from his paper: for every positive integer $k$ there are positive integers $a_1<\cdots<a_k$ with $a_j-a_i\mid(a_i,a_j)$ and $d(a_j)d(a_i/(a_j-a_i))=d(a_i)d(a_j/(a_j-a_i))$ for all $i<j$.

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