Do relaxed Liouville functions violate Chowla's conjecture?

Question

Let $\lambda$ be the Liouville function. One version of Chowla's conjecture says that for each set of distinct natural numbers $h_1 , \dots , h_k$,

$$\sum_{n\leq x} \lambda(n+h_1) \dots \lambda(n+h_k) = o(x)$$

Is this conjectural behavior stable under perturbations?

Say that a function $f: \mathbb N \to \pm 1$ is a relaxed Liouville function with error $\epsilon$ if for each $n$, $f(nm)=\lambda(n)f(m)$ for a density $1-\epsilon$ set of $m$.

Note that this definition is meaningful for $\epsilon = 0$. Then the obvious questions are:

Does there exist a relaxed Liouville function $f$ with error $0$ and a set of numbers $h_1, \dots, h_k$ such that

$$\sum_{n\leq x} f(n+h_1) \dots f(n+h_k) \neq o(x)?$$

and more vaguely:

Does there exist a relaxed Liouville function $f$ with error $\epsilon$ for $\epsilon$ "small", a set of numbers $h_1, \dots, h_k$, and a "large" constant $\delta $ such that:

$$\left|\sum_{n\leq x} f(n+h_1) \dots f(n+h_k)\right| \not \leq ( \delta +o(1) ) x?$$

For instance, can we take $\delta/\epsilon$ to be arbitrarily large with fixed $h_1, \dots, h_k$?

Clearly negative answers to these questions would imply Chowla's conjecture. So I am hoping instead for positive examples.

My motivation is that many elementary approaches that can prove some weak partial results towards Chowla's conjecture are very stable and therefore apply even for relaxed Liouville functions. I want to understand the limits of these elementary methods by explicit counterexample.

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Here is an alternate definition which I think captures the same spirit. Can we find, for some $h_1, \dots, h_k$, for each $N$ a function $f: \mathbb N \to \pm 1$ with $f(pn) = - f(n)$ for $p< N$ such that

$$\left|\sum_{n\leq x} f(n+h_1) \dots f(n+h_k)\right| \not \leq ( \delta +o(1) ) x$$

for some $\delta$ independent of $N$? Or even just some $\delta$ with $\delta \log N$ arbitrarily large?

The idea being that about $1/\log N$ numbers have no prime factors less than $N$ and hence are arbitrary.

Answer 1

If you are willing to violate multiplicativity $f(nm)=f(n)f(m)$ about $\varepsilon$ of the time, then one can make $f$ more or less arbitrary on an interval of the form $[(1-\varepsilon) x, x]$, and this is enough to violate Chowla-type conjectures. So the answer to your question as stated is "yes".

On the other hand, if one retains multiplicativity, then it was conjectured by Elliott that the analog of Chowla's conjecture should hold whenever $\sum_p \frac{1 - \hbox{Re} f(p) \chi(p) p^{it}}{p} = \infty$ for any Dirichlet character $\chi$ and real $t$, which would imply a negative answer to your question for any $\varepsilon < 1/2$ if one insists on perfect multiplicativity. (Technically, Elliott's conjecture is false, see this recent paper of mine with Matomaki and Radziwill, but if one repairs it in the way indicated in that paper, the conclusion of the above argument still holds.) The technical counterexample aside, all other known evidence points to the truth of the (corrected) Elliott conjecture, for instance our paper establishes an averaged form of this conjecture, and a different averaged form was recently established by Frantzikinakis and Host.