The two-point Chowla conjecture (ordinary averages)

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The two-point Chowla conjecture (ordinary averages) parent of this event

Matomaki, Radziwill, Tao, An averaged form of Chowla's conjecture (Algebra Number Theory 2015) evidence for this event

The two-point Chowla conjecture (ordinary averages) evidence for this event

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. parent of this event

Lean proof: OAI/NumberTheory/OrdinaryCorrelations/Elliott/Main.lean evidence for this event

Lean proof: OAI/NumberTheory/TwoPointCorrelations/FinalMain.lean evidence for this event