Shifted Möbius Elliott–Halberstam conjecture

Let $0<\theta<1$ be fixed, let $h\neq0$ be a fixed even integer, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Euler totient function. Shifted Möbius Elliott–Halberstam conjecture. For every $A>0$ and all sufficiently large natural numbers $N$, one has

$$\sum_{q\leq N^{\theta}}\max_{y<N}\max_{(a,q)=1}\left|\sum_{\substack{n\leq y\ \equiv a\,\operatorname{mod} q}}\Lambda(n)\mu(n+h)-\frac{1}{\varphi(q)}\sum_{n\leq y}\Lambda(n)\mu(n+h)\right|\ll_A\frac{N}{\log(N)^A}.$$

This is a Möbius-twisted analogue of Elliott–Halberstam intended to control cancellation on shifted primes. It is used with the classical conjecture in conditional results related to twin primes; the conjecture itself remains open.