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All Problems Random Solved Random Open
Let $f$ be a random $\pm1$-valued multiplicative function, so that 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). Is it true that, almost surely, \[\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=1?\]
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, \[\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{2+o(1)}.\] Harper [Ha13] has shown that the sum is almost surely not $O(N^{1/2}/(\log\log N)^{5/2+o(1)})$.