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Erdős #520 · number theory / probability

Let ff be a Rademacher multiplicative function: a random {−1,0,1}\{-1,0,1\}-valued multiplicative function, where for each prime pp we independently choose f(p)∈{−1,1}f(p)\in \{-1,1\} uniformly at random, and for square-free integers…

MSC 11 · Number theory

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

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let ff be a Rademacher multiplicative function: a random {−1,0,1}\{-1,0,1\}-valued multiplicative function, where for each prime pp we independently choose f(p)∈{−1,1}f(p)\in \{-1,1\} uniformly at random, and for square-free integers nn we extend f(p1⋯pr)=f(p1)⋯f(pr)f(p_1\cdots p_r)=f(p_1)\cdots f(p_r) (and f(n)=0f(n)=0 if nn is not squarefree). Does there exist some constant c>0c>0 such that, almost surely,

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

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Formal statement

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Background

Note that if we drop the multiplicative assumption, and simply assign f(m)=±1f(m)=\pm 1 at random, then this statement is true (with c=2c=\sqrt{2}), the law of the iterated logarithm.

Wintner [Wi44] proved that, almost surely,

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

and Erdős improved the right-hand side to N1/2(log⁡N)O(1)N^{1/2}(\log N)^{O(1)}. Lau, Tenenbaum, and Wu [LTW13] have shown that, almost surely,

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

Caich [Ca24b] has improved this to

∑m≤Nf(m)≪N1/2(log⁡log⁡N)3/4+o(1).\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(N1/2/(log⁡log⁡N)5/2+o(1))O(N^{1/2}/(\log\log N)^{5/2+o(1)}), and conjectured that in fact Erdős' conjecture is false, and almost surely

∑m≤Nf(m)≪N1/2(log⁡log⁡N)1/4+o(1).\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.

Progress & source assessment

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 #520 link
  2. [2]Upstream Lean 4 statement for Erdős #520 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 520 link

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