PROVED
This has been solved in the affirmative.
Let $\epsilon>0$ and $f_\epsilon(p)$ be the smallest integer $m$ such that $\sum_{n\leq N}
\left(\frac{n}{p}\right)<\epsilon N$ for all $N\geq m$. Prove that\[\sum_{p<x}f_\epsilon(p)\sim c_\epsilon \frac{x}{\log x}\]for some $c_\epsilon>0$.
This was proved by Elliott
[El69].
An earlier version of this problem on this site misstated the problem, defining $f_\epsilon(p)$ instead as the smallest integer $m$ such that $\sum_{n\leq m}\binom{n}{p}<\epsilon m$ (thus a 'first-time' problem rather than the 'eventual-time' problem given above). An asymptotic for this alternate definition of $f_\epsilon$ was proved by Tang and Zhang
[TaZh25].
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This page was last edited 27 December 2025. View history
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Additional thanks to: Quanyu Tang
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #981, https://www.erdosproblems.com/981, accessed 2026-04-11
