The two-point Chowla conjecture (ordinary averages)

There is an absolute $c>0$ such that for all fixed integers $a_1,a_2\ge1$, $b_1,b_2\ge0$ with $a_1b_2\ne a_2b_1$, $\left|\sum_{n\le X}\lambda(a_1n+b_1)\lambda(a_2n+b_2)\right|\le C_{a,b}X/(\log X)^c$ for every real $X\ge3$. With $a_1=a_2=1$, $b_1=0$, $b_2=h$ this is the ordinary two-point Chowla conjecture with a power-of-log saving. Constants are ineffective, and coefficients may not grow with $X$. Nothing is proved for three or more points.

There is an absolute $c>0$ such that for all fixed integers $a_1,a_2\ge1$, $b_1,b_2\ge0$ with $a_1b_2\ne a_2b_1$, $\left|\sum_{n\le X}\lambda(a_1n+b_1)\lambda(a_2n+b_2)\right|\le C_{a,b}X/(\log X)^c$ for every real $X\ge3$. With $a_1=a_2=1$, $b_1=0$, $b_2=h$ this is the ordinary two-point Chowla conjecture with a power-of-log saving. Constants are ineffective, and coefficients may not grow with $X$. Nothing is proved for three or more points.