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

Let M\mathcal{M} be the set of multiplicative functions f:NCf:\mathbb{N}\to\mathbb{C} with f(n)1|f(n)|\leq 1. For f,gMf,g\in\mathcal{M}, define

D(f,g;x)=(px1Re(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,NNa_i,b_i,N\in\mathbb{N} satisfying aibjajbia_ib_j\neq a_jb_i whenever iji\neq j, if fMf\in\mathcal{M}' is non-pretentious, then

nxi=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.

Sources & referencesView supporting material

Primary source

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

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.