Problem 520

Erdős Problem #520: declared status 'open'. Formalized: yes. 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,$$\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N\log\log N}}=c?$$ Current best: Wintner [Wi44] proved that, almost surely,$$\sum_{m\leq N}f(m)\ll N^{1/2+o(1)},$$and Erd\H{o}s 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 Erd\H{o}s' conjecture is false, and almost surely$$\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.$$ References [Ca24b] R. Caich, Almost sure upper bound for random multiplicative functions. Prize: no. Tags: number theory, probability.