Elliott's conjecture for correlations of non-pretentious completely multiplicative functions

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Let M\mathcal{M} be the set of multiplicative functions f:N→Cf:\mathbb{N}\to\mathbb{C} with ∣f(n)∣≤1|f(n)|\leq 1. For f,g∈Mf,g\in\mathcal{M}, define

D(f,g;x)=(∑p≤x1−Re⁡(f(p)g(p)‾)p)1/2.\mathbb{D}(f,g;x)=\left(\sum_{p\leq x}\frac{1-\operatorname{Re}(f(p)\overline{g(p)})}{p}\right)^{1/2}.

A function is non-pretentious when it is not at bounded distance from any twisted Dirichlet character χ(n)nit\chi(n)n^{it}. Elliott's conjecture. For fixed ai,bi,N∈Na_i,b_i,N\in\mathbb{N} satisfying aibj≠ajbia_ib_j\neq a_jb_i whenever i≠ji\neq j, if f∈M′f\in\mathcal{M}' is non-pretentious, then

∑n≤x∏i=1Nf(ain+bi)=o(x).\sum_{n\leq x}\prod_{i=1}^{N}f(a_in+b_i)=o(x).

This is presented as an Elliott-type correlation conjecture; the supplied material gives no resolution status, so it remains open.

References

Primary source

Yichen You, “On Completely multiplicative 1 sequences that omit many consecutive +1 values”, arXiv:2404.04981 (2024).

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