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Statement

A positive integer is $S$-smooth when every prime power dividing it is at most $S$ (the note's convention, p. 1).

Lemma 3.3 (p. 6, quoted). "There exist absolute constants $c_1>0$ and $N_1$ such that the following holds for all $N\ge N_1$. Let

$$S:=c_1\frac{N}{(\log N)^3(\log\log N)^3}.$$

Assume that $t_0$ [sic] is $S$-smooth. If $t/3\le s\le t$, then there exists a finite set $A\subseteq[N/16,N]\cap\mathbb N$ such that $\sum_{n\in A}\frac1n=s/t$."

The hypothesis names $t_0$, but the proof and the conclusion concern $t$, so the lemma is read with $t$ the $S$-smooth number. The print does not say that $s$ and $t$ are positive integers; the proof treats them as such (it needs $s/t=x/Q$ with $x$ an integer, which it obtains from $t\mid Q$).

The note introduces the lemma as the strengthened version of Liu and Sawhney's Lemma 4.1 (arXiv:2404.07113v1).

Source. Quanyu Tang, A note on Problem #311, author's note, version 2, dated 15 January 2026; Lemma 3.3 and its proof on p. 6, in Section 3 (pp. 2--6). Not refereed; the edition read and its provenance are recorded in the source digest.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof was read through but rests on Proposition 3.1, whose proof (pp. 3--6) was read for structure only; nothing here is independently reviewed.

Proof pointer

The proof (p. 6) applies Proposition 3.1 (pp. 2--3) with $M=N/16$, $K=10^{-7}N(\log N)^{-1}$ and $S$ as above, for $c_1$ small enough that the proposition's hypotheses on $S,K,M$ hold. A lower bound on the number of smooth integers with few prime factors in $[N/16,N]$ (Liu and Sawhney's Lemma 3.3) makes the reciprocal sum $R$ of the proposition's set of candidates lie between $2$ and $3$, so taking every inclusion probability equal to $(s/t)/R$ puts them in $[1/9,1/2]$ and makes the expected reciprocal sum of the random subset equal to $s/t$. Since $t$ is $S$-smooth it divides $Q$, the least common multiple of the prime powers up to $S$, so $s/t=x/Q$ for an integer $x\in[1,Q]$, and the proposition gives the value $s/t$ with probability at least $1/(4Q)>0$.

Proposition 3.1 refines a special case of Liu and Sawhney's Proposition 3.2; the note's Remark 3.2 (pp. 2--3) lists its four changes to Liu and Sawhney's argument.

Dependencies

Proposition 3.1 of the note; Y. P. Liu and M. Sawhney, arXiv:2404.07113v1 (their Lemma 3.3, and through Proposition 3.1 their Theorem 2.1, Fact 2.5, Lemma 2.6, Lemma 3.1 and the proof of their Proposition 3.2). Used by Theorem 4.1.

Bears on. #311: Theorem 4.1's proof applies the lemma with $t=\mathrm{lcm}(1,\ldots,\lfloor S\rfloor)$ and $s=t-1$ to write $1-1/t$ as a reciprocal sum over $[N/16,N]$, the construction behind that theorem's upper bound for $\delta(N)$; on its own the lemma gives no bound for $\delta(N)$.