Averages of the M\"obius function on shifted primes

TL;DR

This paper proves that the M"obius function exhibits cancellation on shifted primes on average for certain ranges of shifts, extending to prime k-tuples and correlations with other number-theoretic functions.

Contribution

It establishes the conjecture of M"obius cancellation on average for shifts up to H with specific growth conditions, using advanced sieve and correlation techniques.

Findings

Proves M"obius cancellation on average for shifts h ≤ H under certain conditions.

Extends results to shifted prime k-tuples and correlations with von Mangoldt and divisor functions.

Combines sieve methods with refined averaged Chowla's conjecture techniques.

Abstract

It is a folklore conjecture that the M\"obius function exhibits cancellation on 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 since Hildebrand 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 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.