Counting Egyptian fractions

TL;DR

This paper investigates the growth of the number of Egyptian fractions with denominators up to N, providing bounds that involve iterated logarithms, thus advancing understanding of their combinatorial complexity.

Contribution

It establishes new upper and lower bounds for the count of Egyptian fractions with denominators up to N, involving iterated logarithmic functions.

Findings

Lower bound: (N / log N) times product of iterated logs

Upper bound: approximately 0.421 times N

Bounds hold for large N and fixed k ≥ 3

Abstract

For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.

Tables

Table 1. Table 1. The first values of # ​ 𝔈 N # subscript 𝔈 𝑁 \#\mathfrak{E}_{N} .

$N$	${𝔈}_N$	$N$	${𝔈}_N$	$N$	${𝔈}_N$	$N$	${𝔈}_N$
$1$	$2$	$12$	$1856$	$23$	$896512$	$34$	$224129024$
$2$	$4$	$13$	$3712$	$24$	$936832$	$35$	$231010304$
$3$	$8$	$14$	$7424$	$25$	$1873664$	$36$	$237031424$
$4$	$16$	$15$	$9664$	$26$	$3747328$	$37$	$474062848$
$5$	$32$	$16$	$19328$	$27$	$7494656$	$38$	$948125696$
$6$	$52$	$17$	$38656$	$28$	$7771136$	$39$	$1896251392$
$7$	$104$	$18$	$59264$	$29$	$15542272$	$40$	$1928593408$
$8$	$208$	$19$	$118528$	$30$	$15886336$	$41$	$3857186816$
$9$	$416$	$20$	$126976$	$31$	$31772672$	$42$	$3925999616$
$10$	$832$	$21$	$224128$	$32$	$63545344$	$43$	$7851999232$
$11$	$1664$	$22$	$448256$	$33$	$112064512$

Table 2. Table 2. Upper bounds for α 𝛼 \alpha in the case ℓ = 1 ℓ 1 \ell=1 .

$𝔞$	$r$	$\delta /\max \left(𝔞\right)$	$\alpha \le$
$\{1,2,3,6\}$	$13$	$1/12$	$0.67584390$
$\{1,2,3,4,6,12\}$	$29$	$1/28$	$0.66487620$
$\{1,2,3,4,5,6,8,10,12,15\}$	$302$	$1/45$	$0.66601285$
$\{1,2,3,4,5,6,10,12,15,20\}$	$162$	$1/56$	$0.66022083$
$\{1,2,3,4,5,6,8,10,12,15,16,20,24,30\}$	$694$	$1/93$	$0.65915160$
$\{1,2,3,4,5,6,8,9,10,12,15,18,20,24,30\}$	$1061$	$2/195$	$0.65796522$
$\{1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30\}$	$7757$	$7/780$	$0.65533420$

Table 3. Table 3. Upper bounds for α 𝛼 \alpha using increasing collections 𝔞 j := { a 1 , … , a j } assign subscript 𝔞 𝑗 subscript 𝑎 1 … subscript 𝑎 𝑗 \mathfrak{a}_{j}:=\{a_{1},\ldots,a_{j}\} for j = 1 , … , ℓ 𝑗 1 … ℓ j=1,\ldots,\ell and 𝔞 = { d : d | M } 𝔞 conditional-set 𝑑 conditional 𝑑 𝑀 \mathfrak{a}=\{d\colon d|M\} for different M 𝑀 M .

$M$	$\alpha \le$
$5040=2^4\cdot 3^2\cdot 5\cdot 7$	$0.56731289$
$529200=2^4\cdot 3^3\cdot 5^2\cdot 7^2$	$0.55084208$
$55440=2^4\cdot 3^2\cdot 5\cdot 7\cdot 11$	$0.54939823$
$2116800=2^6\cdot 3^3\cdot 5^2\cdot 7^2$	$0.54552567$
$4233600=2^7\cdot 3^3\cdot 5^2\cdot 7^2$	$0.54465996$
$1441440=2^5\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$	$0.53020542$
$2162160=2^4\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13$	$0.52779949$
$4324320=2^5\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13$	$0.52405384$
$43243200=2^6\cdot 3^3\cdot 5^2\cdot 7\cdot 11\cdot 13$	$0.51452256$
$147026880=2^6\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17$	$0.51032288$
$2793510720=2^5\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19$	$0.49944226$
$13967553600=2^6\cdot 3^3\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19$	$0.49153796$
$41902660800=2^6\cdot 3^4\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19$	$0.48948987$

Table 4. Table 4. Upper bounds for α 𝛼 \alpha mixing the computation of the true value of r i subscript 𝑟 𝑖 r_{i} for small divisors, and estimating its value with Lemma  3.3 for large divisors. The relevant parameters are M 𝑀 M and M ′ superscript 𝑀 ′ M^{\prime} .

$M$	$M^{\prime }$	$\alpha \le$
$\mathrm{lcm}\{1,\dots ,17\}$	$\mathrm{lcm}\{1,\dots ,10\}$	$0.52408774$
$\mathrm{lcm}\{1,\dots ,31\}$	$\mathrm{lcm}\{1,\dots ,17\}$	$0.47705355$
$\mathrm{lcm}\{1,\dots ,41\}$	$\mathrm{lcm}\{1,\dots ,17\}$	$0.46441548$
$\mathrm{lcm}\{1,\dots ,53\}$	$\mathrm{lcm}\{1,\dots ,17\}$	$0.44930098$
$\mathrm{lcm}\{1,\dots ,79\}$	$\mathrm{lcm}\{1,\dots ,19\}$	$0.43238142$
$\mathrm{lcm}\{1,\dots ,89\}$	$\mathrm{lcm}\{1,\dots ,19\}$	$0.42691423$
$\mathrm{lcm}\{1,\dots ,97\}$	$\mathrm{lcm}\{1,\dots ,23\}$	$0.42447521$
$\mathrm{lcm}\{1,\dots ,97\}$	$2\cdot 3\cdot \mathrm{lcm}\{1,\dots ,19\}$	$0.42310594$
$\mathrm{lcm}\{1,\dots ,97\}$	$2^2\cdot 3^2\cdot \mathrm{lcm}\{1,\dots ,19\}$	$0.42286665$
$\mathrm{lcm}\{1,\dots ,97\}$	$2^2\cdot 3^2\cdot 5\cdot \mathrm{lcm}\{1,\dots ,19\}$	$0.42099405$

Equations

$$\frac{N}{\log N}\prod _{j=3}^k{\log }_{j}N<\log (\#{\mathfrak{E}}_N)<0.421N,$$

$$\mathfrak{E}_{N}:=\bigg{\{}\sum_{n=1}^{N}\frac{t_{n}}{n}\colon t_{1},\dots,t_{N}\in\{0,1\}\bigg{\}},\qquad N\in\mathbb{N},$$

$${\text{SS}}_N:=\left\{\sum _{n=1}^N\frac{s_n}{n}:s_1,\dots ,s_N\in \{-1,+1\}\right\}\text{and}{\mathfrak{m}}_N(\tau ):=\min \left\{|\tau -\sigma |:\sigma \in {\text{SS}}_N\right\},$$

$$\alpha :=\underset{N\to +\infty }{\mathrm{limsup}}\frac{\log (\#{\mathfrak{E}}_N)}{N},$$

$${\mathfrak{m}}_N(\tau )>\exp \left(-(\alpha +\epsilon )N\right)$$

$$\log (\#{\mathfrak{E}}_N)>\frac{N}{\log N}\prod _{j=3}^k{\log }_{j}N$$

$$\mathcal{E}$$

$$=\bigcap _{M=1}^{\infty }\bigcup _{N\ge M}\{\tau \in \mathbb{R}:{\mathfrak{m}}_N(\tau )\le e^{-(\alpha +\epsilon )N}\}.$$

$$\mathrm{meas}(\mathcal{E})\le \underset{M}{\inf }\sum _{N\ge M}2e^{-(\alpha +\epsilon )N}\#{\text{SS}}_N\le \underset{M}{\inf }\sum _{N\ge M}2e^{-\epsilon N/2}=\underset{M}{\inf }\frac{2e^{-\epsilon M/2}}{1-e^{-\epsilon /2}}=0.$$

$$D:=\{n\in \mathbb{N}:{\mu }_p\mid {\nu }_p(n)\text{ for every prime p}\}$$

$$\delta :=\prod _p\frac{1-p^{-1}}{1-p^{-{\mu }_p}},$$

$$\delta=\Big{(}\sum_{d|M}\frac{1}{d}\Big{)}^{-1}=M\Big{(}\sum_{d|M}d\Big{)}^{-1},$$

$$F(s)$$

$$=\prod _p{\left(1-\frac{1}{p^{{\mu }_{p}s}}\right)}^{-1}=\zeta (s)\prod _p\frac{1-p^{-s}}{1-p^{-{\mu }_{p}s}}=:\zeta (s)H(s).$$

$$\sum _{n\le x}{\delta }_D(n)$$

$$=xH(1)+O\left(x\sum _{d>x}\frac{|h(d)|}{d}\right)+O\left(\sum _{d\le x}|h(d)|\right)=xH(1)+O_{\epsilon }(x^{\epsilon })$$

$$\alpha \le \log 2-\delta \sum _{i=1}^{\ell }\left(\frac{1}{\max ({\mathfrak{a}}_i)}-\frac{1}{\max ({\mathfrak{a}}_{i+1})}\right)\log \left(\frac{2^{\#{\mathfrak{a}}_i}}{r_i}\right),$$

$$\delta:=\Big{(}\sum_{d|M}\frac{1}{d}\Big{)}^{-1}\quad\text{with}\quad M:=\operatorname{lcm}\{a\colon a\in\mathfrak{a}\},$$

$$r_i:=\#R_i\text{with}{R}_i:=\left\{\sum _{a\in {\mathfrak{a}}_i}\frac{t_a}{a}:t_a\in \{0,1\}\right\}.$$

$$D:=\{n\in \mathbb{N}:{\mu }_p\mid {\nu }_p(n)\text{ for every prime p}\}.$$

$${\nu }_p(a_1)-{\nu }_p(a_2)\equiv {\nu }_p(k_2)-{\nu }_p(k_1)\equiv 0(\mathrm{mod}{\mu }_p),$$

$$\delta=\prod_{p}\frac{1-p^{-1}}{1-p^{-\mu_{p}}}=\Big{(}\sum_{d|M}\frac{1}{d}\Big{)}^{-1},$$

$$\#F_N$$

$$=(\delta +o(1))N\sum _{i=1}^{\ell }\left(\frac{1}{\max ({\mathfrak{a}}_i)}-\frac{1}{\max ({\mathfrak{a}}_{i+1})}\right)\#{\mathfrak{a}}_i,$$

$$\sum _{i=1}^{\ell }\sum _{k\in D\cap \left(\frac{N}{k\in \max ({\mathfrak{a}}_{i+1})},\frac{N}{\max ({\mathfrak{a}}_i)}\right]}\sum _{a\in k{\mathfrak{a}}_i}\frac{t_a}{a}+\sum _{b\in F_N^{\prime }}\frac{t_b}{b},$$

$$\log {\mathfrak{E}}_N$$

$$=N\left(\log 2+(\delta +o(1))\sum _{i=1}^{\ell }\left(\frac{1}{\max ({\mathfrak{a}}_i)}-\frac{1}{\max ({\mathfrak{a}}_{i+1})}\right)\log r_i\right)-\#F_N\log 2$$

$$=N\left(\log 2-(\delta +o(1))\sum _{i=1}^{\ell }\left(\frac{1}{\max ({\mathfrak{a}}_i)}-\frac{1}{\max ({\mathfrak{a}}_{i+1})}\right)\log \left(\frac{2^{\#{\mathfrak{a}}_i}}{r_i}\right)\right),$$

$$\alpha \le \log 2-\delta \sum _{i=1}^{\ell }\left(\frac{1}{\max ({\mathfrak{a}}_i)}-\frac{1}{\max ({\mathfrak{a}}_{i+1})}\right)\log \left(\frac{2^{\#{\mathfrak{a}}_i}}{r_i}\right),$$

$$\alpha \le {\alpha }_1:=\log 2-\delta \sum _{i=1}^{\ell }\left(\frac{1}{a_i}-\frac{1}{a_{i+1}}\right)\log \left(\frac{2^i}{r_i}\right),$$

$${\alpha }_1$$

$$\le \log 2-\delta (\log 2-\beta )\sum _{i=1}^{\ell }\frac{1}{i}+O(\delta \log (1+|I|)).$$

$$\delta =\prod _{p\le \ell }\frac{1-p^{-1}}{1-p^{-{\mu }_{p,\ell }}}$$

$$\sum _{p\le \ell }\log (1-p^{-{\mu }_{p,\ell }})=\sum _{p\le \ell }\log (1-{\ell }^{-1})\ll 1/\log (\ell +1),$$

Full text

[TABLE]

for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_{j}x$ denotes the $j$-th iterated logarithm of $x$.

1. Introduction

Every positive rational number $a/b$ can be written in the form of an Egyptian fraction, that is, as a sum of distinct unit fractions: $a/b=1/n_{1}+\cdots+1/n_{r}$ with $n_{1},\dots,n_{r}\in\mathbb{N}$ distinct. Several properties of these representations have been investigated. For example, it is known that all rationals $a/b\in(0,1)$ are representable using only denominators which are $O(b(\log b)^{1+\varepsilon})$ [11, 12] (see also [9]) or that $O(\sqrt{\log b})$ different denominators are always sufficient [10]. It is also well understood which integers can be represented using denominators up to a bound $x$ [4], and Martin [5, 6] showed that any rational can be represented as a “dense Egyptian fraction”. In this paper we take a different direction, and study the cardinality of the set of rational numbers representable using denominators up to $N$,

[TABLE]

as $N\to+\infty$.

Another motivation for studying the cardinality of $\mathfrak{E}_{N}$ comes from the recent work of three of the authors [2] (see also [1]), where the question of how well a real number $\tau$ can be approximated by sums of the form $\sum_{n=1}^{N}s_{n}/n$, where $s_{1},\dots,s_{N}\in\{-1,+1\}$, is studied. Precisely, let

[TABLE]

for every positive integer $N$. Note that $\SS_{N}$ and $\mathfrak{E}_{N}$ have the same cardinality, since $x\mapsto 2x-\sum_{n=1}^{N}1/n$ is a bijection $\mathfrak{E}_{N}\to\SS_{N}$.

It has been proved that \mathfrak{m}_{N}(\tau)<\exp\big{(}-(\tfrac{1}{\log 4}-\varepsilon)(\log N)^{2}\big{)} for every $\tau\in\mathbb{R}$, $\varepsilon>0$, and for all sufficiently large positive integers $N$, depending on $\tau$ and $\varepsilon$ [2, Theorem 1.1]; and that for any $f\colon\mathbb{N}\to\mathbb{R}^{+}$, there exists $\tau_{f}\in\mathbb{R}$ such that $\mathfrak{m}_{N}(\tau_{f})<f(N)$ for infinitely many $N$ [1, Proposition 5.9]. On the other hand, it is possible to obtain lower bounds for $\mathfrak{m}_{N}(\tau)$ holding for almost all $\tau$ by giving upper bounds for the cardinality of $\mathfrak{E}_{N}$. Indeed, defining

[TABLE]

by a Borel-Cantelli argument one obtains the following lower bound.

Lemma 1.1**.**

Fix $\varepsilon>0$. For almost all $\tau\in\mathbb{R}$ we have

[TABLE]

for all sufficiently large $N$, depending on $\tau$ and $\varepsilon$.

In [2, Proposition 2.7] we proved that $\alpha<0.6649$ with a simple argument, improving upon the trivial $\alpha\leq\log 2=0.69314\dots$ In this paper, we improve this bound even further, thus producing a better lower bound for $\mathfrak{m}_{N}(\tau)$.

Theorem 1.2**.**

We have $\alpha<0.421$, that is, $\log(\#\mathfrak{E}_{N})<0.421\,N$ for all $N$ large enough.

In the opposite direction, it is not difficult to show that $\log(\#\mathfrak{E}_{N})\gg N/\log N$. In the following theorem we show that one can slightly improve over this lower bound.

Theorem 1.3**.**

For every integer $k\geq 3$ we have

[TABLE]

for all sufficiently large $N$, depending on $k$.

Theorem 1.3 is proved by showing that the set of integers $N$ for which the quotient $\#\mathfrak{E}_{N}/\#\mathfrak{E}_{N-1}$ attains its maximum value $2$ is quite large. With some more effort it is possible to give an explicit sequence of positive integers $(N_{k})_{k\geq 1}$ such that (1) holds for every $N\geq N_{k}$.

Upper and lower bounds in Theorems 1.2 and 1.3 have different orders and it is not clear whether one of them seizes the true behavior of the sequence $\#\mathfrak{E}_{N}$. Only a few of these numbers can be computed, since the algorithms devised for this purpose have an exponential behavior (in time or in memory). The ones which are known are in Table 1 and partially appear as sequence A072207 of [8]. A graph of $\log(\#\mathfrak{E}_{N})/N$ and of $\log(\#\mathfrak{E}_{N})/(N/\log N)$ is in Figure 1. As the range of $N$ for which $\#\mathfrak{E}_{N}$ is known is very small, the data do not make it clear whether $\log(\#\mathfrak{E}_{N})/N$ might converge to [math] or to any other real number.

Acknowledgements

S. Bettin is member of the INdAM group GNAMPA. L. Grenié, G. Molteni and C. Sanna are members of the INdAM group GNSAGA. C. Sanna is supported by a postdoctoral fellowship of INdAM. The extensive computations needed for this paper have been performed on the UNITECH INDACO computing platform of the Università di Milano and on the computing cluster of the Université de Bordeaux. The authors warmly thank Alessio Alessi and Karim Belabas for their operative assistance for these computations.

Notation

We employ the Landau–Bachmann “Big Oh” notation $O$ as well as the associated Vinogradov symbols $\ll$ and $\gg$. We reserve the letter $p$ for prime numbers. We put $\log_{1}x:=\log x$ and $\log_{k+1}x:=\log(\log_{k}x)$ for every integer $k\geq 1$ and every sufficiently large $x$.

2. Proof of Lemma 1.1

The definition of $\alpha$ implies that we have an upper bound $\#\SS_{N}\leq e^{(\alpha+\varepsilon/2)N}$, for all large enough $N$. The claim follows by the Borel–Cantelli lemma. We have

[TABLE]

Hence, the Lebesgue measure of $\mathcal{E}$ is estimated by

[TABLE]

This implies that, for almost every $\tau$, the lower bound $\mathfrak{m}_{N}(\tau)>e^{-(\alpha+\varepsilon)N}$ holds for all sufficiently large $N$.

3. Proof of Theorem 1.2

For each prime number $p$ and for every positive integer $n$, let $\nu_{p}(n)$ denote the $p$-adic valuation of $n$. We begin with the following easy lemma.

Lemma 3.1**.**

For each prime number $p$, let $\mu_{p}$ be a positive integer, and suppose that $\mu_{p}=1$ for all but finitely many primes $p$. Then, the natural density of the set

[TABLE]

is equal to

[TABLE]

where we note that only finitely many prime numbers contribute to the product. The formula can also be written as

[TABLE]

where $M:=\prod_{p}p^{\mu_{p}-1}$.

Proof.

This is a standard argument, we reproduce it here for completeness. Let $\delta_{D}(n):=1$ when $n\in D$ and $\delta_{D}(n):=0$ otherwise. Note that $\delta_{D}$ is a multiplicative function, so that

[TABLE]

Note that $H(s)$ is a finite Euler product (because $\mu_{p}=1$ for a.e. prime), which is analytic for $\Re(s)>0$. Set $h\colon\mathbb{N}\to\mathbb{R}$, with $H(s)=:\sum_{n=1}^{\infty}h(n)n^{-s}$. Then

[TABLE]

and $\delta=H(1)$. (A more careful analysis shows that the error term has size $\ll(\frac{1}{\ell}\log x)^{\ell}$, uniformly in $\ell\geq 1$ and $x\geq 2$.) The alternative representation of $\delta$ as a sum of divisors of $M$ follows immediately by the unique factorization of integers as a product of prime powers. ∎

The next lemma is our key tool to provide numerical upper bounds for $\alpha$.

Lemma 3.2**.**

Let $\mathfrak{a}_{1}\subset\cdots\subset\mathfrak{a}_{\ell}=:\mathfrak{a}$ be finite nonempty sets of natural numbers. We have

[TABLE]

where $1/\max(\mathfrak{a}_{\ell+1}):=0$,

[TABLE]

and, for $i=1,\dots,\ell$,

[TABLE]

Proof.

For any any prime $p$ let $\mu_{p}:=1+\max\{\nu_{p}(a)\colon a\in\mathfrak{a}\}$ and

[TABLE]

Let $k\mathfrak{a}:=\{ka\colon a\in\mathfrak{a}\}$ for every integer $k$. Note that $k_{1}\mathfrak{a}\cap k_{2}\mathfrak{a}=\varnothing$ for all $k_{1},k_{2}\in D$ with $k_{1}\neq k_{2}$. Indeed, suppose that $k_{1}a_{1}=k_{2}a_{2}$ for some $k_{1},k_{2}\in D$ and $a_{1},a_{2}\in\mathfrak{a}$. Then, for every prime number $p$, we have

[TABLE]

which in turn implies that $\nu_{p}(a_{1})=\nu_{p}(a_{2})$. Hence, $a_{1}=a_{2}$ and $k_{1}=k_{2}$.

Clearly $\mu_{p}$ is $1$ for all but finitely many prime numbers $p$. Hence, Lemma 3.1 applies, and the natural density of the set $D$ is

[TABLE]

with $M=\prod_{p}p^{\mu_{p}-1}=\operatorname{lcm}\{a\colon a\in\mathfrak{a}\}$. Let $N$ be a positive integer. Since $\mathfrak{a}_{1},\dots,\mathfrak{a}_{\ell}$ are contained in $\mathfrak{a}$, we have that the sets $k\mathfrak{a}_{i}$, with $i\in\{1,\ldots,\ell\}$ and $k\in D\cap\left(N/\max(\mathfrak{a}_{i+1}),N/\max(\mathfrak{a}_{i})\right]$ are pairwise disjoint. Let $F_{N}$ be their union. Clearly, $F_{N}\subseteq\{1,\dots,N\}$. Also, we have

[TABLE]

as $N\to+\infty$.

We are finally ready to give an upper bound for $\#\mathfrak{E}_{N}$. Every element of $\mathfrak{E}_{N}$ is of the form

[TABLE]

where $t_{n}\in\{0,1\}$ for all $n\in\{1,\dots,N\}$, and $F_{N}^{\prime}:=\{1,\dots,N\}\setminus F_{N}$. Therefore, we have

[TABLE]

as $N\to+\infty$, where we used (2). Consequently,

[TABLE]

as claimed. ∎

Lemma 3.2 gives non-trivial bounds for $\alpha$ already when applied in the simplest case $\ell=1$. Table 2 displays, for some choices of $\mathfrak{a}=\mathfrak{a}_{1}$, the value of the parameter $r=r_{1}$ (obtained numerically) and the corresponding bound $\alpha\leq\log 2-\frac{\delta}{\max(\mathfrak{a})}\log({2^{\#\mathfrak{a}}}/{r})$.

Notice that it is always convenient to include $1$ in $\mathfrak{a}$ since its presence does not affect the fraction $\delta/\max(\mathfrak{a})$ whereas it typically increases the value of $\log({2^{\#\mathfrak{a}}}/{r})$. The result in [2, Proposition 2.7] corresponds to the above construction with $\mathfrak{a}=\{1,2,3,4,6,12\}$. A judicious choice of a larger $\mathfrak{a}$ improves the result, but the gain in $\log({2^{\#\mathfrak{a}}}/{r})$ is considerably tempered by the size of $\delta/\max(\mathfrak{a})$, which becomes smaller and smaller. The last entry in Table 2 represents the best bound we were able to obtain with this approach.

Better results can be attained by taking $\ell>1$. Given $1\leq a_{1}<a_{2}<\cdots<a_{\ell}$, one can take the collection $\mathfrak{a}_{1}\subset\mathfrak{a}_{2}\subset\ldots\subset\mathfrak{a}_{\ell}=:\mathfrak{a}$ with $\mathfrak{a}_{i}=\{a_{1},\ldots,a_{i}\}$ for $i=1,\dots,\ell$. In this case the bound in Lemma 3.2 simplifies to

[TABLE]

with $1/a_{\ell+1}:=0$. A first naive choice is to take $\mathfrak{a}_{i}:=\{1,\ldots,i\}$ for $1\leq i\leq\ell$. Notice that with this choice $r_{i}=\#\mathfrak{E}_{i}$. Now, for some $I\leq\ell$ let $\beta>0$ be such that $r_{i}\leq e^{\beta i}$ for $I\leq i\leq\ell$. Then

[TABLE]

Now

[TABLE]

with $\mu_{p,\ell}=1+\max\{t\colon p^{t}\leq\ell\}=1+[\frac{\log\ell}{\log p}]\geq\frac{\log\ell}{\log p}.$ Thus, by Merten’s theorem and since

[TABLE]

one obtains $\delta\sim e^{-\gamma}/\log\ell$ as $\ell\to\infty$. In particular, taking for example $I=[e^{\sqrt{\log\ell}}]$, we get

[TABLE]

as $\ell\to\infty$ and so for any $\varepsilon>0$ we obtain $\alpha_{1}<\log 2-e^{-\gamma}(\log 2-\beta)+\varepsilon$ if $\ell$ is large enough. By the definition of $\alpha$ we have that for all $\varepsilon>0$ one has $\beta\leq\alpha+\varepsilon$ for large enough $\ell$, thus giving $\alpha_{1}<\log 2-e^{-\gamma}(\log 2-\alpha-\varepsilon)$. Notice that due to the loss of the factor $e^{-\gamma}$ this argument fails to recover an upper bound of the type $\alpha+\varepsilon$ for $\alpha_{1}$, and for example it would only gives $\alpha_{1}\leq(1-e^{-\gamma})\log 2+\varepsilon=0.3039\dots+\varepsilon$ if $\alpha=0$. However, it shows that by taking $\ell$ large enough one can surely get arbitrarily close to the upper bound $\log 2-e^{-\gamma}(\log 2-\alpha)$ just by performing a finite numerical computation.

An alternative and more effective approach arises from the observation that the density $\delta$ depends only on the valuations $\mu_{p}$, and so it is the same for different $\mathfrak{a}$ sharing the same least common multiple. In particular, it is convenient to select the numbers in $\mathfrak{a}$ as the full collection of divisors of a given integer $M$. In this case the bound further simplifies to

[TABLE]

Indeed, if $a_{1}<\cdots<a_{\ell}$ are all the divisors of $M$, then we get

[TABLE]

Numerically it seems that best results come from taking $M$ with only “small” prime divisors (say, for example, $M=\operatorname{lcm}\{1,\dots,m\}$, for some $m$, or some small variations of it) and this could be explained by the observation that with numbers of this shape an argument analogous to the above does not have a loss of $e^{-\gamma}$ (notice however that in this case $r_{i}$ is not equal to $\#\mathfrak{E}_{i}$, so the argument is not fully recursive as in the previous case). Table 3 collects some bounds one gets in this way. Most likely further improvements could be obtained by taking larger numbers. However, the quantity of time and memory one needs to compute $r_{i}$ for large $i$ prevented us to any significant improvement on the value appearing at the bottom of Table 3.

The search of a good way to store the huge vectors containing the numbers in $R_{i}$ led us to consider the following upper bound for $r_{i}$.

Lemma 3.3**.**

Let $\mathfrak{a}:=\{a_{1}<a_{2}<\cdots<a_{\ell}\}$ be the full set of divisors of an integer $M$, let $\mathfrak{a}_{i}:=\{a_{1},\ldots,a_{i}\}$, and let $R_{i}$ and $r_{i}$ as in Lemma 3.2. Then

[TABLE]

Proof.

Let $L_{i}:=\operatorname{lcm}\{a_{1},\dots,a_{i}\}$. Hence, $L_{i}/a_{k}\in\mathbb{Z}$ for $k=1,\dots,i$. Consequently, $L_{i}R_{i}=\{\sum_{k=1}^{i}\pm L_{i}/a_{k}\}\subseteq\mathbb{Z}$. Moreover, $\max\{L_{i}R_{i}\}=-\min\{L_{i}R_{i}\}=M_{i}:=\sum_{k=1}^{i}L_{i}/a_{k}$, so that $\#(L_{i}R_{i})\leq 1+2M_{i}$. We can improve this bound by a factor $2$ since $\pm 1\equiv 1\pmod{2}$, so that all numbers in $L_{i}R_{i}$ have the same parity of $M_{i}$. As a consequence, $r_{i}=\#R_{i}=\#(L_{i}R_{i})\leq 1+M_{i}$, as desired. ∎

At this point, we are ready to explain the method that we used to prove the bound of Theorem 1.2. The main idea is to use the upper bound given by (3), but computing the exact value of $r_{i}$ only for small $i$. Precisely, let $M$ be a (very large) positive integer and let $a_{1}<\cdots<a_{\ell}$ be all its divisors. Also, let $M^{\prime}$ be a (small) divisor of $M$ and let $a_{1}^{\prime}<\cdots<a_{m}^{\prime}$ be all its divisors. For each divisor $a_{j}^{\prime}$ of $M^{\prime}$ we pre-compute the value $r_{j}^{\prime}:=\left\{\sum_{a\in\mathfrak{a}_{j}^{\prime}}\frac{t_{a}}{a}:t_{a}\in\{0,1\}\right\}$, where $\mathfrak{a}_{j}^{\prime}:=\{a_{1}^{\prime},\dots,a_{j}^{\prime}\}$. Then, for each divisor $a_{i}$ of $M$ we look for the largest $a_{j}^{\prime}$ dividing $a_{i}$, and we estimate $r_{i}$ with the minimum between what we get from Lemma 3.3 and the number $r_{j}^{\prime}\cdot 2^{\#\mathfrak{a}_{i}-\#\mathfrak{a}_{j}^{\prime}}$. This is a correct bound since, by the assumption on $j$, we have $\mathfrak{a}_{j}^{\prime}\subseteq\mathfrak{a}_{i}$ and consequently $r_{i}$ is at most $r_{j}^{\prime}$ multiplied by the power of two elevated to the difference of the cardinalities of $\mathfrak{a}_{i}$ and $\mathfrak{a}_{j}^{\prime}$.

Table 4 collects some results we get in this way for suitable choices of $M^{\prime}$ and $M$. The last entry gives our best result, which proves Theorem 1.2. Its computation needed approximatively 40 hours and 200GB.

4. Proof of Theorem 1.3

Define the set

[TABLE]

and let $\mathcal{U}(x):=\mathcal{U}\cap[1,x]$ for every $x\geq 1$.

Lemma 4.1**.**

If $N\in\mathcal{U}$ then $\#\mathfrak{E}_{N}=2\#\mathfrak{E}_{N-1}$. Consequently, $\#\mathfrak{E}_{N}\geq 2^{\#\mathcal{U}(N)}$.

Proof.

On the one hand, $N\in\mathcal{U}$ implies that $\mathfrak{E}_{N-1}\cap(\mathfrak{E}_{N-1}+1/N)=\varnothing$. On the other hand, $\mathfrak{E}_{N}=\mathfrak{E}_{N-1}\cup(\mathfrak{E}_{N-1}+1/N)$. Hence, we have $\#\mathfrak{E}_{N}=\#\mathfrak{E}_{N-1}+\#(\mathfrak{E}_{N-1}+1/N)=2\#\mathfrak{E}_{N-1}$, as claimed. ∎

In light of Lemma 4.1, to produce a lower bound for $\#\mathfrak{E}_{N}$ it is sufficient to give a lower bound for $\#\mathcal{U}(N)$.

Lemma 4.2**.**

$\mathcal{U}$* contains $1$ and all prime numbers.*

Proof.

The fact that $1\in\mathcal{U}$ follows immediately from the definition of $\mathcal{U}$. Furthermore, for every prime number $p$ and for every $w_{1},\dots,w_{p-1}\in\{-1,0,+1\}$, the identity

[TABLE]

is impossible, since the left-hand side has nonnegative $p$-adic valuation (all denominators are not divisible by $p$) while the right-hand side has negative $p$-adic valuation. Hence, $p\in\mathcal{U}$. ∎

For each positive integer $m$, let $d_{m}:=\operatorname{lcm}\{1,\dots,m\}$ and $g_{m}:=d_{m}\sum_{j=1}^{m}1/j$.

Lemma 4.3**.**

If $m\in\mathcal{U}$ and $p>g_{m}$ is a prime number, then $mp\in\mathcal{U}$.

Proof.

Suppose by contradiction that $N:=mp\notin\mathcal{U}$. Hence, there exist $w_{1},\dots,w_{N-1}\in\{-1,0,+1\}$ such that

[TABLE]

Consequently, splitting the sum according to whether $p|n$ and multiplying by $p$ we obtain

[TABLE]

which in turn implies that

[TABLE]

Since $m\in\mathcal{U}$, the number on the left hand side is non-zero. Moreover, the absolute value of its numerator is at most $g_{m}$, which by hypothesis is strictly smaller than $p$. But then (4) is impossible. ∎

Remark 4.1*.*

The proofs of Lemma 4.2 and Lemma 4.4 can be easily adapted to show that if $m\in\mathcal{U}$ and $p>g_{m}$ is a prime number then $mp^{k}\in\mathcal{U}$ for every positive integer $k$. However, this generalization does not lead to an improvement of our final result.

Lemma 4.4**.**

If $m\in\mathcal{U}$ and $p>3^{m}$ is a prime number, then $mp\in\mathcal{U}$.

Proof.

In light of Lemma 4.3, it is enough to prove that $g_{m}<3^{m}$ for every positive integer $m$. On the one hand, from [7, p. 228] we know that $d_{m}=\exp(\psi(m))\leq\exp(1.04m)$. On the other hand,

[TABLE]

Hence, $g_{m}\leq\exp(1.04m)\log(em)<3^{m}$ for every integer $m\geq 25$. A direct computation shows that the $g_{m}<3^{m}$ holds also for $m=1,\dots,24$. ∎

As usual, let $\pi(x)$ denote the number of prime numbers not exceeding $x$. The next lemma gives a recursive lower bound for $\#\mathcal{U}(x)$.

Lemma 4.5**.**

Let $y\geq 1$ and $x\geq 3^{y}$. Then

[TABLE]

Proof.

Let us consider the natural numbers of the form $mp$, where $p$ is a prime number satisfying $3^{y}<p\leq x/m$ and $m\in\mathcal{U}(y)$. Thanks to Lemma 4.4, we have that $mp\in\mathcal{U}(x)$. Moreover, these numbers can be written in the form $mp$ in a unique way. Indeed, for the sake of contradiction, suppose that $mp=m^{\prime}p^{\prime}$ for some $m^{\prime},p^{\prime}$ satisfying the same conditions as $m,p$, with $p\neq p^{\prime}$. Then, $p^{\prime}\mid m$, so that $3^{y}<p^{\prime}\leq m\leq y$, which is impossible. At this point, counting the choices for $m$ and $p$, we get

[TABLE]

The desired claim follows by applying the inequality $\pi(3^{y})y\leq 2\cdot 3^{y}$, which in turn follows from the estimate $\pi(x)\leq 2x/\log x$ valid for all $x\geq 2$. ∎

We need the following technical lemma. Let $e_{1}:=1$ and $e_{k+1}:=e^{e_{k}}$ for all integers $k\geq 2$.

Lemma 4.6**.**

For any fixed integer $k\geq 2$, we have

[TABLE]

as $x\to+\infty$.

Proof.

For $k=2$ the empty product appearing on the left hand side is set to $1$, by definition, and the claim is clear in this case. Assume $k\geq 3$. We have

[TABLE]

as $x\to+\infty$. The claim then follows from de l’Hôpital’s rule. ∎

Lemma 4.7**.**

For every integer $k\geq 2$, we have

[TABLE]

for all sufficiently large $x$, depending on $k$.

Proof.

We proceed by induction on $k$. By Lemma 4.2 and by Chebyshev’s estimate, we have

[TABLE]

for all sufficiently large $x$. This proves the claim for $k=2$. Suppose $k\geq 3$ and that we have already proved the claim for $k-1$. Put $y:=\tfrac{1}{2}\log x$ and assume that $x$ is sufficiently large. By Lemma 4.5 and by Chebyshev’s estimate, we have

[TABLE]

Furthermore, by partial summation and by the induction hypothesis, we get

[TABLE]

for all sufficiently large $x$, depending on $k$, where we employed Lemma 4.6. Putting together (5) and (6) we obtain the desired claim. ∎

Theorem 1.3 follows easily by Lemma 4.1 and 4.7.