OPEN
This is open, and cannot be resolved with a finite computation.
Let $f$ be a Rademacher multiplicative function: a random $\{-1,0,1\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\cdots p_r)=f(p_1)\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely,\[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?\]
Note that if we drop the multiplicative assumption, and simply assign $f(m)=\pm 1$ at random, then this statement is true (with $c=\sqrt{2}$), the
law of the iterated logarithm.
Wintner
[Wi44] proved that, almost surely,\[\sum_{m\leq N}f(m)\ll N^{1/2+o(1)},\]and Erdős improved the right-hand side to $N^{1/2}(\log N)^{O(1)}$. Lau, Tenenbaum, and Wu
[LTW13] have shown that, almost surely,\[\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{2+o(1)}.\]Caich
[Ca24b] has improved this to\[\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{3/4+o(1)}.\]Harper
[Ha13] has shown that the sum is almost surely not $O(N^{1/2}/(\log\log N)^{5/2+o(1)})$, and conjectured that in fact Erdős' conjecture is false, and almost surely\[\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.\]
View the LaTeX source
Additional thanks to: Mehtaab Sawhney
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 #520, https://www.erdosproblems.com/520, accessed 2026-01-14
