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, $\sum_{p{\,\leqslant} X}\mu(p+h) \ = \ o(\pi(X))$ 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 $h{\,\leqslant} H$, 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.

On the Hardy–Littlewood–Chowla conjecture on average

“…This improves on work of the first author [10] that established Corollary 1.4 when for any function tending to infinity with X .…”

“…Theorem 1.6 improves on the range , which follows (under a slightly different pretentiousness hypothesis) from Fourier uniformity bounds of Matomäki, Radziwiłł, Tao, Ziegler and the second author [15, Theorem 1.8] (see [10, Theorem 1.8] for the details of this implication) 2 . We also note that the case of Conjecture 1.5 was proven on average by Matomäki–Radziwiłł–Tao [12] in the regime .…”

“…It is likely that with some extra effort (in particular, refining Proposition 2.1 for in the spirit of [10, Theorem 2.2]), Theorem 1.2(ii) could be extended to the regime . However, our main interest here is in taking H as small as possible.…”

“…The proof method of Theorem 1.2 (and Theorem 1.6) is somewhat different from that of [10] (or [12]). Both proofs begin with Fourier-analytic identities, namely, [10, Lemma 2.1] and Proposition 3.2 below.…”

Autocorrelation of the Mobius Function

“…This pair of finite sums over the shifted primes is a topic of current research, see [11] for extensive details on recent developments.…”