It is a folklore conjecture that the M\"obius function exhibits cancellation
on shifted primes; that is, βpβ€XβΞΌ(p+h)Β =Β o(Ο(X)) as
Xββ for any fixed shift h>0. We prove the conjecture on average for
shifts hβ€H, provided logH/loglogXββ. We also obtain results
for shifts of prime k-tuples, and for higher correlations of M\"obius with
von Mangoldt and divisor functions. Our argument combines sieve methods with a
refinement of Matom\"aki, Radziwi\l\l, and Tao's work on an averaged form of
Chowla's conjecture.Comment: 24 page