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Claim. Let $g(n)=\#\{m\ge1:\phi(m)=n\}$. Theorem 1.1 of the preprint: for a fixed nonzero integer $a$ and every $\beta>15/(32\sqrt e)=0.2843\ldots$, at least $x/(\log x)^{C}$ primes $p\in(x,2x]$ have every prime factor of $p-a$ at most $x^{\beta}$. Corollary 1.3, by the Erdős--Pomerance transfer: the integers $m$ with at least $m^{0.7156}$ solutions $n$ of $\phi(n)=m$ form an infinite sequence $m_1<m_2<\cdots$ with $\log m_{i+1}/\log m_i\to1$. So

$$g(n)\ge n^{0.7156}$$

for infinitely many $n$, which answers the question of Problem 821 for every $\epsilon\ge0.2844$, since then $n^{1-\epsilon}\le n^{0.7156}$. The technical input is Theorem 1.4, a mean value theorem for primes in arithmetic progressions to quadrilinear moduli $qrst$ reaching moduli up to $x^{17/32-\epsilon}$, beyond Maynard's $x^{11/21}$. The statements are on the card Lichtman 2022; the proof is not reconstructed in this repository. The fiber exponent $1-15/(32\sqrt e)=0.7156\ldots$ improves the $0.7039$ of Baker and Harman (Baker and Harman 1998).

Covers. The range $\epsilon\ge0.2844$. Corollary 1.3 gives infinitely many $n$ with $g(n)\ge n^{0.7156}$, a non-strict inequality; the strict one the problem asks for follows from Theorem 1.1, which holds for every $\beta>15/(32\sqrt e)=0.28434\ldots$: the same transfer at a $\beta$ in $(15/(32\sqrt e),0.2844)$ gives infinitely many $n$ with $g(n)\ge n^{1-\beta}>n^{0.7156}\ge n^{1-\epsilon}$. The question for smaller $\epsilon$, and so the problem as posed, is not addressed.

Depends on. Nothing in this wiki.

Standing. Claimed. The arXiv record of arXiv:2211.09641 has one version, submitted 14 November 2022, which dates this page, and lists no journal reference; no journal record of the paper is known, so no refereed evidence is listed. The site's commentary (page last edited 1 October 2025) gives $g(n)>n^{0.71568\cdots}$ infinitely often as the best known bound and credits it to Lichtman, but labels the problem OPEN, which is not an acceptance, so no reviewed evidence is listed. The formal-conjectures statement file for the problem states the bound as erdos_821.variants.lichtman (category research solved) with a sorry body at the commit the problem page links; it is not a formalization.