Problem Statement
Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C$ tending to infinity such that, for every $\lambda>0$, the integral\[\int_C \lvert f(z)\rvert^{-\lambda} \mathrm{d}z\]is finite?
Categories:
Analysis
Progress
Huber [Hu57] proved that for every $\lambda>0$ there is such a path $C_\lambda$ such that this integral is finite.This is true. The case when $f$ has finite order was proved by Zhang [Zh77]. The general case was proved by Lewis, Rossi, and Weitsman [LRW84], who in fact proved this with $\lvert f\rvert$ replaced by $e^u$ where $u$ is any subharmonic function.
Source: erdosproblems.com/515 | Last verified: January 15, 2026