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It is a folklore conjecture that the M\"obius function exhibits cancellationon shifted primes; that 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 sinceHildebrand in 1989. We prove the conjecture on average for shifts $h\le H$,provided $\log H/\log\log X\to\infty$. We also obtain results for shifts ofprime $k$-tuples, and for higher correlations of M\"obius with von Mangoldt anddivisor functions. Our argument combines sieve methods with a refinement ofMatom\"aki, Radziwi\l\l, and Tao's work on an averaged form of Chowla'sconjecture.

Averages of the M\"obius function on shifted primes