Vanishing correlations for large sets of primes

Determine whether, for every large subset \(P\) of the primes and every fixed finite set \(H=\{h_1,\ldots,h_d\}\) of non-negative integers, the limit \(\lim_{x\to\infty}x^{-1}\sum_{n\leq x}\lambda_P(n+h_1)\cdots\lambda_P(n+h_d)\) exists and equals zero.

Background

The paper proves an Euler-product formula for the correlation limit associated with λP\lambda_P when PP is a small set of primes, meaning that the sum of the reciprocals of the primes in PP converges. It also proves that nontrivial cancellation occurs whenever the relevant limit exists and PP is non-empty.

The case of large prime sets is not covered by the main theorem. The authors expect the correlation limit to vanish in that setting; establishing this would extend the main theorem to all prime subsets and, in particular, would imply Chowla’s conjecture when PP is the set of all primes, so that λP\lambda_P is the Liouville function.

References

Theorem \ref{Theorem "Main Theorem"} still leaves open the question of large subsets of primes. In this case, we expect the limit on the left hand side of Equation "Chowla" to vanish, thus making Theorem \ref{Theorem "Main Theorem"} true in this case as well (if we interpret the infinite product on the right of Equation "Chowla" as zero, similar to the infinite product on the right hand side of Equation "WW").

Equation "Chowla":

κPH:=lim⁡x→∞1x∑n⩽xΛPH(n)=∏p∈P(1−2ηpH).\kappa_P^H:=\lim_{x\to\infty}\frac{1}{x} \sum_{n\leqslant x} \Lambda_P^H(n) = \prod_{p\in P} \left(1 - 2\eta_p^H\right).

Equation "WW":

lim⁡x→∞1x∑n⩽xλP(n)=∏p∈P(1−2p+1).\lim_{x\to\infty} \frac{1}{x} \sum_{n\leqslant x} \lambda_P(n) = \prod_{p\in P} \left(1-\frac{2}{p+1}\right).

— On variants of Chowla's conjecture  (2501.10962 - Krishnamoorthy, 19 Jan 2025) in Section 1, Introduction, immediately following Theorem 1 (Main Theorem)