Yokota 2002 number integers representable sums unit fractions iii

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corollary_1: The bounds the citing problems consume: for large n the number of integers that are sums of reciprocals of distinct integers at most n is at least log n + γ − (π²/3 + o(1))(log log n)²/log n, improving Croot's constant 9/2, and every large integer a is such a sum with denominators at most exp[a − γ + (π²/3 + o(1))(log a)²/a]; Croot's upper bound is quoted alongside.

theorem_1: Yokota's main theorem: for large n, the integers that are sums of reciprocals of distinct integers at most n drawn from the divisor set D(t) number at least log n + γ − (π²/3 + o(1))(log log n)²/log n, and every large integer a is such a sum with denominators at most exp[a − γ + (π²/3 + o(1))(log a)²/a].

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H. Yokota, On the Number of Integers Representable as Sums of Unit Fractions, III, Journal of Number Theory 96 (2002), no. 2, 351--372, doi:10.1006/jnth.2002.2797 (both printed on p. 351, with the copyright line "2002 Elsevier Science (USA)"); the author at the Department of Mathematics, Hiroshima Institute of Technology; communicated by D. Goss; received July 16, 2001, revised January 4, 2002 (p. 351). Cited as [Yo02] on the problem pages. It is the third paper of a series: the paper's references 10 and 11 are Yokota, On number of integers representable as sums of unit fractions, II, J. Number Theory 67 (1997), 162--169, and its Corrigendum, J. Number Theory 72 (1998), 150 (the problem pages' [Yo97], filed as yokota_1997_number_integers_representable_sum_unit_fractions_ii; the Corrigendum is not held); its reference 2 is Croot's Mathematika 46 (1999) paper, filed as crootiii_1999_questions_erdos_graham_about_egyptian_fractions; its reference 5 is the 1980 Erdős--Graham monograph, cited for pp. 30--44, filed as erdos_1980_old_new_problems_results_combinatorial_number_theory; and its reference 1 is Bleicher and Erdős, Denominators of Egyptian fractions, II, filed as bleicher_1976_denominators_egyptian_fractions_ii.

The copy read for this card is the publisher's production PDF of the journal article: 22 pages, printed pp. 351--372 = PDF pp. 1--22 (printed p. $n$ is PDF p. $n-350$), typeset from the publisher's composition system (3B2 and Acrobat Distiller 4.05 per the file's metadata, created 28 September 2002), with a text layer that reads the prose cleanly and garbles the mathematics (parentheses, inequality signs, minus signs, Greek letters and the product and sum signs come out as substitute characters, so every display was read on the page image). Provenance: the copy was obtained on 2026-09-22 as a free copy from the publisher's open archive, the DOI https://doi.org/10.1006/jnth.2002.2797 resolving to the article's PDF on the publisher's site (PII S0022314X02927976) under the publisher's user license; 215,556 bytes. The PDF prints "© 2002 Elsevier Science (USA)" and, on the next line, "All rights reserved." on its first page (printed p. 351), every other right reserved.

Read status: claims checked for the abstract, the definition of $N(n)$, the recalled bounds of the author's earlier papers and of Croot (p. 351), the definitions of $F(a)$, the sequence $S$, $p_{k(t)}$, $p_{u(t)}$, $D(t)$, $L(t)$, $N^*(n)$ and $F^*(a)$, and Theorem 1 (p. 352), Corollary 1 and the statements of Lemmas 1--5 (p. 353), each read clause by clause on the page images of PDF pp. 1--3 on 2026-09-22; the closing step of the proof of Theorem 1 and the opening of § 4 with the proof of Lemma 1 (p. 357) were read on the page image of PDF p. 7, and the reference list (pp. 371--372) on the page images of PDF pp. 21--22. The proof of Theorem 1 (pp. 354--357) and the proofs of Lemmas 2--5 (pp. 357--371) were read in the text layer for structure only, and none of their estimates was checked; the results cited in the proof of Lemma 2 on p. 360 were read on the page image of PDF p. 10 on 2026-10-07. No proof was checked, and nothing here is independently reviewed.

Contents

• Abstract and § 1, Introduction (pp. 351--353, page images). "Let $N(n)$ be the set of all integers that can be expressed as a sum of reciprocals of distinct integers $\le n$. Then we prove that for sufficiently large $n$, $\log n+\gamma-(\frac{\pi^2}3+o(1))\frac{(\log_2n)^2}{\log n}\le|N(n)|$, which improves the lower bound given by Croot." Here $\log_j$ is the $j$-fold iterated logarithm (p. 351), so $\log_2n=\log\log n$. The introduction writes $N(n)$ as the set of integers $a=\sum_{1\le k\le n}\varepsilon_k/k$ with $\varepsilon_k\in\{0,1\}$, recalls that the author's earlier papers [10--13] showed $\log n+\gamma-2-o(1)\le|N(n)|\le\log n+\gamma-(\frac14+o(1))\frac{(\log_2n)^2}{\log n}$ in answer to questions of Erdős and Graham [5], and that Croot [2] improved this to $\log n+\gamma-(\frac92+o(1))\frac{(\log_2n)^2}{\log n}\le|N(n)|\le\log n+\gamma-(\frac12+o(1))\frac{(\log_2n)^2}{\log n}$. Page 352 records a remark and question of Don Zagier, put to the author in private communication: with $F(a)=\min\{n:a\in N(n)\}$, "determining $N(n)$ for all $n$ is the same as calculating $F(a)$ since $a\in N(n)$ iff $n\ge F(a)$", and Zagier asked whether the upper bound of $F(a)$ can be improved. The construction: $S=\{s_j\}$ is the increasing sequence of all positive integers of the form $p^{2^i}$, $p$ prime, $i\ge0$; for a chosen $s_t$, $p_{k(t)}$ is the largest prime $<s_t^2$ and $p_{u(t)}$ the smallest prime $>s_t$; $D(t)=\{d\le L(t):d\mid\prod_1^ts_i\prod_{u(t)}^{k(t)}p_j\}$ with $L(t)=p_{k(t)}(\log p_{k(t)})^2(\log_2p_{k(t)})^2$; $N^*(n)$ is the set of integers $a=\sum_{k\in D(t),k\le n}\varepsilon_k/k$ and $F^*(a)=\min\{n:a\in N^*(n)\}$, so that $N^*(n)\subset N(n)$, $|N^*(n)|\le|N(n)|$ and $F(a)\le F^*(a)$. Theorem 1 (p. 352) and Corollary 1 (p. 353) are quoted on their result pages; the corollary's upper bound is Croot's, quoted.
• § 2, Lemmata (p. 353, page image). Five lemmas, stated without proof. Lemma 1: for all large $p_{k(t)}$, $p_{k(t-1)}\ge p_{k(t)}(1-2^{2/5}/p_{k(t)}^{1/5})$. Lemma 2: for all $t\ge t_0$ and all $r=(1+o(1))\frac{(\log_2p_k)^2}{\log p_k}\prod_1^ts_i$ there are distinct integers $d_i$ with $r=\sum d_i$, $d_i\mid\prod_1^ts_i$ and $d_i\ge\prod_1^ts_i/p_k\log p_k\,(1-\frac2{\log_2p_k})$. Lemma 3: the same for $r=(1+o(1))\frac{(\log_2p_k)^2}{\log p_k}\prod_1^ts_i\prod_u^kp_j$ with $d_i\mid\prod_1^ts_i\prod_u^kp_j$ and $d_i\ge\prod_1^ts_i\prod_u^kp_j/L(t)$. Lemma 4: for $t\ge t_0$, $(\frac{15-\pi^2}3+o(1))\frac{(\log_2p_{k(t)})^2}{\log p_{k(t)}}\le\sum_{d\le L(t),d\notin D(t)}\frac1d\le(\frac{\pi^2-3}3+o(1))\frac{(\log_2p_{k(t)})^2}{\log p_{k(t)}}$. Lemma 5: for $t\ge t_0$, $\sum_{d\le L(t),d\in D(t)}\frac1d-\sum_{d\le L(t-1),d\in D(t-1)}\frac1d\le(4+o(1))\frac{\log_2p_{k(t)}}{\log^2p_{k(t)}}$.
• § 3, Proof of Theorem 1 (pp. 354--357; p. 357 on the page image, the rest in the text layer). Given a large integer $a$, $t$ is chosen so that $a+[(\log a)^2]/a$ lies between the reciprocal sums over $D(t-1)$ up to $L(t-1)$ and over $D(t)$ up to $L(t)$; Lemmas 4 and 5 place $a$ between $\log L(t)+\gamma-(\frac{\pi^2}3+o(1))\frac{(\log_2p_{k(t)})^2}{\log p_{k(t)}}$ (display (1), p. 355) and $\log L(t)+\gamma-(\frac{18-\pi^2}3+o(1))\frac{(\log_2p_{k(t)})^2}{\log p_{k(t)}}$ (display (2)), so $a=(1+o(1))\log p_{k(t)}$. The deficit $\sum_{d\in D(t),d\le L(t)}1/d-a$ is written as $r^*/\prod_1^ts_i\prod_{u(t)}^{k(t)}p_j$ with $r^*=(1+o(1))\frac{(\log_2p_{k(t)})^2}{\log p_{k(t)}}\prod_1^ts_i\prod_{u(t)}^{k(t)}p_j$, Lemma 3 expresses $r^*$ as a sum of distinct divisors $d_i$ with $\prod s_i\prod p_j/d_i\le L(t)$, and removing those reciprocals leaves $a=\sum_{j\le L(t),j\in D(t)}\varepsilon_j/j$. Hence $a\in N^*(L(t))$, $F^*(a)\le L(t)\le\exp[a-\gamma+(\frac{\pi^2}3+o(1))\frac{(\log a)^2}a]$ (p. 356), and for $L(t)\le n<L(t+1)$ Lemma 1 gives $\log L(t+1)=\log L(t)+O(p_{k(t)}^{-1/5})$, so $\log n+\gamma-(\frac{\pi^2}3+o(1))\frac{(\log_2n)^2}{\log n}\le|N^*(n)|$ (p. 357).
• § 4, Proof of Lemmas (pp. 357--371; the proof of Lemma 1 on the page image of p. 357, the rest in the text layer). Lemma 1 (p. 357) from the prime-gap bound $p_{i+1}\le p_i+p_i^{3/5}$ of Heath-Brown and Iwaniec [4]. Lemma 2 (pp. 357--361) splits into the cases $s_t=p^{2^l}$ with $l\ge1$ and $s_t=p$, builds complete residue systems modulo $s_t$ from divisor sets $D_j$ by Lorentz's theorem [6] and the Cauchy--Davenport theorem [3], cites Lemma 1 of the author's 1991 paper [9], and on p. 360 uses Lemma 1 of [8] and Lemma 2 of [1]. Lemma 3 (pp. 361--364) uses Lemma 1 of the author's 1988 paper [8], Lemma 2 of Bleicher and Erdős [1] and Theorem 2.2 of [9]. Lemma 4 (pp. 364--370) estimates the reciprocal sum over $d\le L(t)$ outside $D(t)$, using the prime-sum estimates of Rosser and Schoenfeld [7]. Lemma 5 (pp. 370--371) splits the difference of the two reciprocal sums into three sums $S_1$, $S_2$, $S_3$ and bounds each, the middle one giving the $(4+o(1))\log_2p_{k(t)}/\log^2p_{k(t)}$.
• References (pp. 371--372, page images), thirteen items: Bleicher and Erdős, Denominators of Egyptian fractions, II (Illinois J. Math. 20, 1976); Croot III, On some question of Erdős and Graham about Egyptian fractions (Mathematika 46, 1999); Davenport, On the addition of residue classes (1935); Heath-Brown and Iwaniec, On the difference between consecutive primes (Invent. Math. 55, 1979); Erdős and Graham, Old and New Problems and Results in Combinatorial Number Theory, pp. 30--44 (1980); Lorentz, On a problem of additive number theory (1954); Rosser and Schoenfeld, Approximate formula for some functions of prime numbers (1962); and the author's papers Denominators of Egyptian fractions (J. Number Theory 28, 1988), On a problem of Erdős and Graham (J. Number Theory 39, 1991), On number of integers representable as sums of unit fractions, II (J. Number Theory 67, 1997) with its Corrigendum (J. Number Theory 72, 1998), and The largest integer expressible as a sum of reciprocal of integers (J. Number Theory 76, 1999) with its Erratum (J. Number Theory 83, 2000).

Compiled scope

The paper is compiled at statement depth for the results the citing problems consume: Theorem 1 (p. 352) and Corollary 1 (p. 353), read on the page images and paged on theorem_1 and corollary_1. The lemmas are recorded as statements read on the page image; the proofs were read for structure only, and nothing is independently reviewed.

Bears on. #309: Corollary 1 (printed p. 353, PDF p. 3) is the lower bound the site's commentary attributes to the paper: "There exists a constant $n_0$ such that, for all $n>n_0$, $\log n+\gamma-(\frac{\pi^2}3+o(1))\frac{(\log_2n)^2}{\log n}\le|N(n)|\le\log n+\gamma-(\frac12+o(1))\frac{(\log_2n)^2}{\log n}$", where $|N(n)|$ counts the empty sum $0$, so $|N(n)|=F(N)+1$ for the problem's $F(N)$, and the upper bound is Croot's, quoted; the paper's own contribution is the lower bound, proved in Theorem 1 (p. 352) for the subset $N^*(n)$ of representations with denominators in $D(t)$, so the constant $\frac92$ of Croot's lower bound becomes $\frac{\pi^2}3$. For the integer $F(N)$ the lower bound can hold only in the integer-part form of Croot's Main Theorem, not read as the site words it (see the corollary's page). The corollary was read on the page image at statement depth; the proof was read for structure only. #308: the second display of Corollary 1 (p. 353), "$F(a)\le F^*(a)\le\exp[a-\gamma+(\frac{\pi^2}3+o(1))\frac{(\log a)^2}a]$" with $F(a)=\min\{n:a\in N(n)\}$ (p. 352), is the inverse form of the smallest-missing-integer question: it sharpens the constant $\frac92$ of Croot's Corollary, and, by the deduction written on the corollary's result page (not stated in the paper), gives $\{1,\ldots,\lfloor H_N-(\frac{\pi^2}3+o(1))(\log\log N)^2/\log N\rfloor\}\subseteq N(N)$ for large $N$, which lowers the upper threshold of Croot's two-case window from $\frac92$ to $\frac{\pi^2}3$ without closing it.

Results.

• Theorem 1 (p. 352): for all $n>n_0$, $\log n+\gamma-(\frac{\pi^2}3+o(1))\frac{(\log_2n)^2}{\log n}\le|N^*(n)|$ and $F^*(a)\le\exp[a-\gamma+(\frac{\pi^2}3+o(1))\frac{(\log a)^2}a]$.
• Corollary 1 (p. 353): the same bounds for $|N(n)|$ and $F(a)$, with Croot's upper bound for $|N(n)|$ quoted alongside.

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