OPEN
This is open, and cannot be resolved with a finite computation.
Let $f$ be a random completely multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random. Is it true that\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N}}=\infty\]with probability $1$?
This model of a random multiplicative function is sometimes called a Rademacher function, although this is sometimes reserved for a merely multiplicative function (which is $0$ on non-squarefree integers). See
[520] for the partial sums of this alternative model.
It should also be compared to another popular model of random completely multiplicative functions, Steinhaus functions, which have $f(p)$ uniformly distributed over the unit circle.
Atherfold
[At25] has proved that, almost surely,\[\sum_{m\leq N}f(m)\ll N^{1/2}(\log N)^{1+o(1)}.\]
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This page was last edited 26 January 2026.
Additional thanks to: Adam Harper
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T. F. Bloom, Erdős Problem #1144, https://www.erdosproblems.com/1144, accessed 2026-03-01
