Erdős's totient-fiber conjecture: $\varphi(m)=n$ has more than $n^{1-\varepsilon}$ solutions for infinitely many $n$

Lineage and corrections

Theorem 1.1: for every $\varepsilon>0$ there are infinitely many $n$ with $g(n)>n^{1-\varepsilon}$, which is optimal in the exponent because $g(n)\ll_\eta n^{1+\eta}$. The input is Theorem 1.2: for fixed $0<\delta<1/4$, at least $x^{1-o(1)}$ primes $p\in(2x,5x]$ have $P^+(p-1)\le x^\delta$; in particular there are infinitely many primes with $P^+(p-1)\le p^\varepsilon$ for every $\varepsilon>0$. It does not give a positive proportion of such primes (that stronger statement is in the Poisson-Dirichlet companion), and the $o(1)$ is not made explicit. parent of this event

Erdős Problem #821 (erdosproblems.com) evidence for this event

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VibeMathed record: Erdős's totient-fiber conjecture: $\varphi(m)=n$ has more than $n^{1-\varepsilon}$ solutions for infinitely many $n$ evidence for this event

Companion: Weighted dilation graphs, smooth shifted primes and totient fibers evidence for this event

Erdős's totient-fiber conjecture: $\varphi(m)=n$ has more than $n^{1-\varepsilon}$ solutions for infinitely many $n$ parent of this event

The Ford-Konyagin-Luca conjecture: the prime factors of $p-1$ follow the Poisson-Dirichlet law evidence for this event

Companion: Prime predecessors with an even number of prime factors evidence for this event