../

---

Source. Scott D. Hughes, Sums of distinct divisors of factorials, arXiv:2609.10902v1, Theorem 2 (quoted from Berend–Harmse), display (1) and Corollary 3 with its proof, physical p. 2 of the five-page PDF held by Hughes (2026); the library records them on the Theorem 2 page. Read in the canonical conversion beside the PDF and checked against the page image. Consumed by the Theorem 1 reconstruction.

Standing. Author-recorded reconstruction; not an independent review; it changes no status and assigns no tier. The Berend–Harmse estimate is an imported theorem, quoted below in the form the source prints; the 1993 paper is not held and was not read, so the import is second-hand.

Definitions

$\log$ is the natural logarithm and $\lg t=\log_2t$. For real $x\ge2^{16}$ put

$$\varepsilon_x=\Bigl(\frac1x\Bigr)^{\frac{\lg x}2-\lg(\lg x)} .$$

Expanding $\lg x=\log x/\log2$ and $\lg(\lg x)=\log(\log x/\log2)/\log2$,

$$\log\frac1{\varepsilon_x} =\Bigl(\frac{\lg x}2-\lg(\lg x)\Bigr)\log x =\frac{(\log x)^2}{2\log2}\Bigl(1-\frac{2\log(\log x/\log2)}{\log x}\Bigr). \tag{1}$$

The window of index $j\ge2$ is the interval $[\sqrt{(j-1)!},\sqrt{j!}]$; its logarithmic width is $\tfrac12\log j$.

Imported theorem (Berend–Harmse, as quoted)

D. Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 3, 569–583, Theorem 2, in the form the source prints: for every integer $n\ge2^{16}$ and every real $D$ with $\sqrt{(n-1)!}\le D\le\sqrt{n!}$ there is a divisor $x$ of $n!$ with

$$\Bigl|\frac xD-1\Bigr| \le5\cdot10^7\Bigl(\frac{\lg n}n\Bigr)^{\frac{\lg n-\lg(\lg n)+1}2+\lg e} \le\Bigl(\frac1n\Bigr)^{\frac{\lg n}2-\lg(\lg n)}=\varepsilon_n .$$

Only the outer inequality $|x/D-1|\le\varepsilon_n$ is consumed. The second inequality between the two bounds is a numerical comparison, which was checked here: the base-two logarithm of the ratio of the left bound to the right one equals

$$\lg(5\cdot10^7)-\Bigl(\tfrac12+\lg e\Bigr)\lg n-\tfrac12(\lg\lg n)^2 +\Bigl(\tfrac12+\lg e\Bigr)\lg\lg n ,$$

which is about $-5.7$ at $n=2^{16}$ and decreases in $n$.

Two facts about the error term

Monotonicity. Write $L=\log x$. The exponent in (1) is

$$\Bigl(\frac{\lg x}2-\lg(\lg x)\Bigr)\log x =\frac1{\log2}\Bigl(\frac{L^2}2-L\log\frac L{\log2}\Bigr),$$

whose derivative in $L$ is $\frac1{\log2}\bigl(L-\log(L/\log2)-1\bigr)$. At $L=16\log2$ this is positive, since $16\log2-\log16-1>0$, and it increases with $L$. So $\log(1/\varepsilon_x)$ increases for $x\ge2^{16}$, and the sequence $(\varepsilon_j)_{j\ge2^{16}}$ is decreasing.

Size. At $j=2^{16}$ the exponent is $\tfrac{16}2-\lg16=4$, so $\varepsilon_{2^{16}}=2^{-64}$, and therefore $\varepsilon_j\le2^{-64}<\tfrac14$ for every integer $j\ge2^{16}$.

Statement

Let $j\ge2^{16}$ and $n\ge j$ be integers, and let $a<b$ be consecutive divisors of $n!$ with $\sqrt{(j-1)!}\le\sqrt{ab}\le\sqrt{j!}$. Then

$$\log\frac ba\le3\varepsilon_j .$$

Proof

Apply the imported theorem with $j$ in place of $n$ and $D=\sqrt{ab}$, which lies in the required range. It gives a divisor $x$ of $j!$ with $|x/D-1|\le\varepsilon_j$. Since $j\le n$, $j!$ divides $n!$, so $x$ is a divisor of $n!$; as $a<b$ are consecutive divisors of $n!$, no divisor of $n!$ lies strictly between them, so $x\le a$ or $x\ge b$.

If $x\le a$, then $1-\varepsilon_j\le x/D\le a/D=\sqrt{a/b}$, so $b/a\le(1-\varepsilon_j)^{-2}$ and $\log(b/a)\le-2\log(1-\varepsilon_j)$.

If $x\ge b$, then $\sqrt{b/a}=b/D\le x/D\le1+\varepsilon_j$, so $b/a\le(1+\varepsilon_j)^2$ and $\log(b/a)\le2\log(1+\varepsilon_j)\le2\varepsilon_j$.

For $0\le u\le\tfrac14$,

$$-\log(1-u)=\log\Bigl(1+\frac u{1-u}\Bigr)\le\frac u{1-u}\le\frac43u .$$

With $u=\varepsilon_j<\tfrac14$ the first case gives $\log(b/a)\le\tfrac83\varepsilon_j$. Both cases give $\log(b/a)\le3\varepsilon_j$.