Hardy–Littlewood–Elliott conjecture

Let $h_1,\ldots,h_k,a_1,\ldots,a_\ell$ be distinct integers, and let $f_1,\ldots,f_k:\mathbb{N}\to\mathbb{C}$ be 1-bounded multiplicative functions. Define the pretentious distance by

$$\mathbb{D}(f,g;X)=\left(\sum_{p\leq X}\frac{1-\operatorname{Re}(f(p)\overline{g(p)})}{p}\right)^{1/2}$$

and

$$M(f;X,Q)=\inf_{\substack{|t|\leq X\chi\,(q),\ q\leq Q}}\mathbb{D}\bigl(f,n\mapsto n^{it}\chi(n);X\bigr)^2.$$

Assume that $f_1$ is non-pretentious, meaning that $M(f_1;X,Q)\to\infty$ as $X\to\infty$ for every $Q\geq1$.

Hardy–Littlewood–Elliott conjecture. Then

$$\sum_{n\leq X}f_1(n+h_1)\cdots f_k(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_\ell)=o(X).$$

This generalizes the Hardy–Littlewood–Chowla framework from the Möbius function to arbitrary 1-bounded multiplicative functions. The case $\ell=0$ is identified in the source with Elliott’s conjecture on correlations of multiplicative functions; the general assertion remains open.