The Chowla conjecture asserts, among other things, that one has the asymptotic
as for any distinct integers , where is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms than the shifts , but for sake of discussion let us focus on the shift case.) This conjecture remains open for , though there are now some partial results when one averages either in or in the , as discussed in this recent post.
A natural generalisation of the Chowla conjecture is the Elliott conjecture. Its original formulation was basically as follows: one had
for all Dirichlet characters and real numbers . It is easy to see that some condition like (2) is necessary; for instance if and has period then can be verified to be bounded away from zero as .
for some that was sufficiently large depending on , and all Dirichlet characters of period at most . As further support of this conjecture, I recently established the bound
under the same hypotheses, where is an arbitrarily slowly growing function of .
In view of these results, it is tempting to conjecture that the condition (4) for one of the should be sufficient to obtain the bound
when is large enough depending on . This may well be the case for . However, the purpose of this blog post is to record a simple counterexample for . Let’s take for simplicity. Let be a quantity much larger than but much smaller than (e.g. ), and set
For , Taylor expansion gives
On the other hand one can easily verify that all of the obey (4) (the restriction there prevents from getting anywhere close to ). So it seems the correct non-asymptotic version of the Elliott conjecture is the following:
Conjecture 1 (Non-asymptotic Elliott conjecture) Let be a natural number, and let be integers. Let , let be sufficiently large depending on , and let be sufficiently large depending on . Let be bounded multiplicative functions such that for some , one has
for all Dirichlet characters of conductor at most . Then
The case of this conjecture follows from the work of Halasz; in my recent paper a logarithmically averaged version of the case of this conjecture is established. The requirement to take to be as large as does not emerge in the averaged Elliott conjecture in my previous paper with Matomaki and Radziwill; it thus seems that this averaging has concealed some of the subtler features of the Elliott conjecture. (However, this subtlety does not seem to affect the asymptotic version of the conjecture formulated in that paper, in which the hypothesis is of the form (3), and the conclusion is of the form (1).)
In my previous paper with Matomaki and Radziwill, we could show that easier expression
for some large . However, to obtain an analogous bound for (5) it now appears that one needs to strengthen the above condition to
in order to address the counterexample in which for some between and . This seems to suggest that proving (5) (which is closely related to the case of the Chowla conjecture) could in fact be rather difficult; the estimation of (6) relied primarily of prior work of Matomaki and Radziwill which used the hypothesis (7), but as this hypothesis is not sufficient to conclude (5), some additional input must also be used.