Averages of the Möbius Function on Shifted Primes

It is a folklore conjecture that the M\"obius function exhibits cancellation on shifted primes; is, $\sum_{p\le X}\mu(p+h) \ = o(\pi(X))$ as $X\to\infty$ for any fixed shift $h>0$. This appears in print at least since Hildebrand 1989. We prove average shifts $h\le H$, provided $\log H/\log\log X\to\infty$. also obtain results of prime $k$-tuples, and higher correlations with von Mangoldt divisor functions. Our argument combines sieve methods refinement Matom\"aki, Radziwi\l\l, Tao's work an averaged form Chowla's conjecture.