On the Hardy--Littlewood--Chowla conjecture

An old conjecture dating back at least to Hildebrand’s work in 1989, asserts that the Möbius function exhibits cancellation on shifted primes; that is, $$\sum _{p\le x}\mu (p+2)=o(\pi (x)),\text{or equivalently}\sum _{n\le x}\mathrm{\Lambda }(n)\mu (n+2)=o(x)\mathrm{.}$$ This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the two sets $\{p:\mu (p+2)=1\}$ and $\{p:\mu (p+2)=-1\}$ has asymptotic density $1\mathrm{/}2$ among the primes $p$ with ${\mu }^2(p+2)=1$: \begin{equation*} \lim_{x\to\infty} \frac{#{p\le x:\ μ(p+2)=κ}} {#{p\le x:\ μ^2(p+2)=1}} =\frac12, \quad κ\in{-1,,1}. \end{equation*}