Liouville cancellation along shifted primes

Prove that the Liouville function has zero average along shifted primes, namely that lim_{N→∞} π(N)^{-1}∑_{p≤N} λ(p+1)=0.

Background

The paper presents cancellation of the Liouville function along the sequence p+1, with p ranging over primes, as an analogue of the prime number theorem for the shifted-prime subsequence. Although an averaged form of higher-order shifted-prime Chowla correlations is cited as known, the individual one-point assertion is described as a folklore conjecture.

References

It is a folklore conjecture that \begin{equation}\label{pnt_shifted_primes} \lim_{N\to\infty} \frac1{\pi(N)}\sum_{p\leq N}\lambda(p+1)=0, \end{equation}

pnt_shifted_primes:

lim⁡N→∞1π(N)∑p≤Nλ(p+1)=0,\lim_{N\to\infty} \frac1{\pi(N)}\sum_{p\leq N}\lambda(p+1)=0,

— A dynamical generalization of Chowla's conjecture on average  (2608.16108 - Wang, 17 Aug 2026) in Section 1, Introduction and statement of results