Averages of the Möbius function on shifted primes

Abstract

It is a folklore conjecture that the Möbius 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 Möbius with von Mangoldt and divisor functions. Our argument combines sieve methods with a refinement of Matomäki, Radziwiłł, and Tao's work on an averaged form of Chowla's conjecture.

25 pages

References in corpus (1)

• A note on the Liouville function in short intervals

Cited by in corpus (2)

• The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero
• On the Hardy-Littlewood-Chowla conjecture on average