Problem Statement
Let $f(z)=\sum_{k=1}^\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\geq 1$). Is it true that if $n_k/k\to \infty$ then $f(z)$ assumes every value infinitely often?
Categories:
Analysis
Progress
A conjecture of Fejér and Pólya.Fejér [Fe08] proved that if $\sum\frac{1}{n_k}<\infty$ then $f(z)$ assumes every value at least once, and Biernacki [Bi28] proved that if $\sum\frac{1}{n_k}<\infty$ then $f(z)$ assumes every value infinitely often.
Pólya [Po29] proved that if $f$ has finite order then $f(z)$ assumes every value infinitely often under the assumption that $\limsup (n_{k+1}-n_k)=\infty$.
Source: erdosproblems.com/517 | Last verified: January 15, 2026