Sarnak's conjecture

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2. Sign patterns

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Problem 2.1.
Let $f$ be a multiplicative function with countable range. Can we find a coloring of that range with (say) two colors, such that the number of "sign patterns" of $f$ (in terms of the colors) grows super-linearly?
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Problem 2.2.
Can we show arbitrarily large patterns $++\ldots+$ or $--\ldots -$ in $(\lambda(n))_{n \in \N}$ ? An approach through the entropy decrement argument (roughly) reduces this to showing the existence of arbitrarily large progressions $n, n+p, \dots, n+kp$ with $\lambda(n) = \lambda(n+p) = \dots = \lambda(n+kp)$ .

Cite this as: AimPL: Sarnak's conjecture, available at http://aimpl.org/sarnakconjecture.