PROVED
This has been solved in the affirmative.
- $500
If $f:\mathbb{N}\to \{-1,+1\}$ then is it true that for every $C>0$ there exist $d,m\geq 1$ such that\[\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?\]
The Erdős discrepancy problem. This is true, and was proved by Tao
[Ta16], who also proved the more general case when $f$ takes values on the unit sphere.
In several places (e.g.
[Er64b],
[Er65b], and
[Er81]) Erdős further conjectured that\[\max_{md\leq x}\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert \gg \log x.\]In
[Er85c] Erdős also asks about the special case when $f$ is multiplicative.
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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 #67, https://www.erdosproblems.com/67, accessed 2026-01-16
