Problem 821

Erdős Problem #821: declared status 'open'. Formalized: no. Let $g(n)$ count the number of $m$ such that $\phi(m)=n$. Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that$$g(n) > n^{1-\epsilon}?$$ Current best: The best known bound is that there are infinitely many $n$ such that$$g(n) > n^{0.71568\cdots},$$obtained by Lichtman [Li22] as a consequence of proving that there are $\geq \frac{x}{(\log x)^{O(1)}}$ many primes $p\leq x$ such that all prime factors of $p-1$ are $\leq x^{0.2843\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman [BaHa98]). Prize: no. OEIS: A014197. Tags: number theory.