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OP-9369OpenNumber theory

Erdős #1144 · number theory / probability

Let ff be a random completely multiplicative function, where for each prime pp we independently choose f(p)∈{−1,1}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}…
MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Let ff be a random completely multiplicative function, where for each prime pp we independently choose f(p)∈{−1,1}f(p)\in \{-1,1\} uniformly at random. Is it true that

lim sup⁡N→∞∑m≤Nf(m)N=∞\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N}}=\infty

with probability 11?

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Background

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 00 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)f(p) uniformly distributed over the unit circle.

Atherfold [At25] has proved that, almost surely,

∑m≤Nf(m)≪N1/2(log⁡N)1+o(1).\sum_{m\leq N}f(m)\ll N^{1/2}(\log N)^{1+o(1)}.

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1144 link

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