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Statement

$f_3(m,n)$ is the number of solutions $a_1\le a_2\le a_3$ in positive integers of $m/n=1/a_1+1/a_2+1/a_3$ (p. 2).

Theorem 3 (pp. 3--4). "For given $m\in\mathbb N$ there are infinitely many $n\in\mathbb N$ such that

$$f_3(m,n)\ \ge\ \exp\Bigl((\log6+o_m(1))\frac{\log n}{\log\log n}\Bigr).$$

Furthermore, for given $m\in\mathbb N$, there exists a subset $\mathcal M_1$ of the integers with density one, such that for any $n\in\mathcal M_1$

$$f_3(m,n)\ \ge\ \Bigl(\frac1{\varphi(m)}+o(1)\Bigr)\exp\bigl((\log3+o_m(1))\log\log n\bigr)\cdot\log\log n \ \gg\ (\log n)^{\log3+o_m(1)}.$$

For the special case $m=4$ and for integers $n$ in a set $\mathcal M_2\subset\mathbb N$ with density one, the last bound may be improved to

$$f_3(4,n)\ \ge\ \exp\bigl((\log6+o(1))\log\log n\bigr).$$

"

The paper compares these with Elsholtz and Tao's Theorem 1.8 (infinitely many $n$ with $f_3(4,n)\ge\exp((\log3+o(1))\log n/\log\log n)$ and a density-one set with $f_3(4,n)\ge\exp((\tfrac{\log3}2+o(1))\log\log n)$), noting $\log3=1.09861\ldots$, $\tfrac{\log3}2=0.54930\ldots$ and $\log6=1.79175\ldots$ (p. 3).

Source. Elsholtz and Planitzer, arXiv:1805.02945v1 (8 May 2018); Theorem 3 on pp. 3--4, read on the page images. Published as Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1401--1427, DOI 10.1017/prm.2018.137; the published version was not compared.

Read depth. Claims checked: the statement was read clause by clause on the page images of pp. 3--4. The proof (Section 7, pp. 16--18) was later read for its structure only and was not checked step by step. Remark 1 (p. 4) says the improvement comes from using factorizations of many shifts of $n$, not of $n$ alone.

Proof pointer

For the first bound the proof (pp. 16--17) takes $n=mn'$ with $n'$ the product of the first $r$ primes and counts solutions of $1/n'=1/a_1+1/a_2+1/a_3$ with $a_1=n'+d$ for a divisor $d$ of $n'$, splitting the remainder by a pair of coprime divisors; this gives $3^{\omega(n')}$ pairs for each of the $2^{\omega(n')}$ choices of $d$. The density-one bounds (p. 17 for $\mathcal M_1$, pp. 17--18 for $\mathcal M_2$) build the first denominator from a prime divisor of $n'$ in the residue class $-n'\bmod m'$, where $m/n=m'/n'$ in lowest terms (for $\mathcal M_1$), or, for $m=4$, from a divisor of $n/4$, of $n/2$, or of $n$ in the class $-n\bmod4$, according to $n\bmod4$ (for $\mathcal M_2$), using the Turán--Kubilius inequality and a result on divisors in residue classes (the paper's reference [17, Theorem 5]). Remark 3 (p. 18) explains the gap between the constants $\log3$ and $\log6$ for general $m$, and says the exponent $\log6$ can be achieved for a set of density one within the integers coprime to $m$.

Dependencies

The prime number theorem (through $\omega(n')\sim\log n'/\log\log n'$), the Turán--Kubilius inequality and the paper's reference [17, Theorem 5]; not examined here.

Bears on

• Problem 242: the site's "for almost all $n$, $f(n)\ge(\log n)^{\log6+o(1)}$" is the case $m=4$ of the last bound ($\exp((\log6+o(1))\log\log n)=(\log n)^{\log6+o(1)}$); $f_3$ counts nondecreasing triples, so repeated denominators are not excluded; a lower bound on a density-one set says nothing about the remaining $n$.