Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper “An averaged form of Chowla’s conjecture“. This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that

as for any distinct natural numbers , where denotes the Liouville function. (One could also replace the Liouville function here by the Möbius function and obtain a morally equivalent conjecture.) This conjecture remains open for any ; for instance the assertion

is a variant of the twin prime conjecture (though possibly a tiny bit easier to prove), and is subject to the notorious parity barrier (as discussed in this previous post).

Our main result asserts, roughly speaking, that Chowla’s conjecture can be established unconditionally provided one has non-trivial averaging in the parameters. More precisely, one has

Theorem 1 (Chowla on the average)Suppose is a quantity that goes to infinity as (but it can go to infinity arbitrarily slowly). Then for any fixed , we haveIn fact, we can remove one of the averaging parameters and obtain

Actually we can make the decay rate a bit more quantitative, gaining about over the trivial bound. The key case is ; while the unaveraged Chowla conjecture becomes more difficult as increases, the averaged Chowla conjecture does not increase in difficulty due to the increasing amount of averaging for larger , and we end up deducing the higher case of the conjecture from the case by an elementary argument.

The proof of the theorem proceeds as follows. By exploiting the Fourier-analytic identity

(related to a standard Fourier-analytic identity for the Gowers norm) it turns out that the case of the above theorem can basically be derived from an estimate of the form

uniformly for all . For “major arc” , close to a rational for small , we can establish this bound from a generalisation of a recent result of Matomaki and Radziwill (discussed in this previous post) on averages of multiplicative functions in short intervals. For “minor arc” , we can proceed instead from an argument of Katai and Bourgain-Sarnak-Ziegler (discussed in this previous post).

The argument also extends to other bounded multiplicative functions than the Liouville function. Chowla’s conjecture was generalised by Elliott, who roughly speaking conjectured that the copies of in Chowla’s conjecture could be replaced by arbitrary bounded multiplicative functions as long as these functions were far from a twisted Dirichlet character in the sense that

(This type of distance is incidentally now a fundamental notion in the Granville-Soundararajan “pretentious” approach to multiplicative number theory.) During our work on this project, we found that Elliott’s conjecture is not quite true as stated due to a technicality: one can cook up a bounded multiplicative function which behaves like on scales for some going to infinity and some slowly varying , and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations will be large). In our paper we propose a technical “fix” to Elliott’s conjecture (replacing (1) by a truncated variant), and show that this repaired version of Elliott’s conjecture is true on the average in much the same way that Chowla’s conjecture is. (If one restricts attention to real-valued multiplicative functions, then this technical issue does not show up, basically because one can assume without loss of generality that in this case; we discuss this fact in an appendix to the paper.)

## 28 comments

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17 March, 2015 at 11:39 pm

AnonymousIs there any (probabilistic) prediction for the true size of the term in theorem 1 ?

18 March, 2015 at 12:16 pm

Terence TaoStandard probabilistic heuristics suggest that this expression should be of size for any .

18 March, 2015 at 5:50 am

Joel MoreiraThis looks impressive! I am wondering: does this averaged version imply anything about Sarnak’s conjecture on Mobius orthogonality?

18 March, 2015 at 12:18 pm

Terence TaoNot as far as we can tell – there’s just too much averaging. We had initially hoped that we could say something nontrivial about a weaker version of Sarnak’s conjecture, namely that the Liouville sequence itself had positive entropy (i.e. the word growth is exponential) but even showing it is faster than linear remains a challenge.

18 March, 2015 at 3:01 pm

AnonymousWhat is known about the “irreducible polynomial version” of this kind of problem (average of for all polynomials of degree on for going to infinity, “Hardy-Littlewood conjecture” on the number of irreducibles of degree such that is also irreducible, etc.)?

18 March, 2015 at 6:36 pm

Joel MoreiraI am not sure if this is what you are looking for, but an analogue of the Chowla conjecture (with the “right” decay) has been proved for polynomials over finite fields by Carmon and Rudnick: http://arxiv.org/abs/1205.1599.

19 March, 2015 at 10:47 am

AnonymousThank you very much for this reference. It gives good estimates for fixed and going to infinity: I am wondering if there are estimates for fixed and going to infinity (which for me seems more similar to the case of the natural numbers).

19 March, 2015 at 3:48 pm

bradYou’re right that such estimates would be more similar to the natural numbers. Unfortunately, I don’t think there are any such bounds at the moment. Roughly speaking, the paper of Carmon and Rudnick proceeds by reducing Moebius sums to N-fold character sums over , and then uses a Weil bound to estimate 1 part of this sum non-trivially, and the remaining N-1 parts trivially. It seems like it would require a really different idea to non-trivially bound several parts of the sum together. On the other hand, its worth remarking that the Galois-theoretic Pellet formula (1.5) holds true whether q tends to infinity or N does, and is an additional bit of information that has no analogue to be exploited over the integers…

19 March, 2015 at 4:27 pm

bradThis looks like a nice paper! Is there any hope one could use an extension of these methods to show the existence of some fixed such that ? More loosely, could it be possible to give some control on how the set of in Theorem 1.2 varies with ?

19 March, 2015 at 9:43 pm

Terence TaoUnfortunately we don’t have any mechanism to make these correlations for different x “talk” to each other. Perhaps the best that could be done is to find a sequence of going to infinity and a sequence of going to infinity such that goes to zero as , but this is quite a weak statement.

19 March, 2015 at 10:07 pm

AnonymousIs it possible to generalize theorem 1 by considering powers of the inner sum (i.e. estimating its moments)?

19 March, 2015 at 10:57 pm

Terence TaoYes, but one arrives at an equivalent estimate: for instance, since , the first estimate in Theorem 1 implies that for any fixed . (The converse implication can also be recovered from Holder’s inequality.)

25 March, 2015 at 6:43 am

AnonymousAre the implied constants in theorem 1.6 (in the arxiv paper) effective?

25 March, 2015 at 8:11 am

Terence TaoYes; we make no use of exceptional zeroes or similar repulsion phenomena, which are the main sources of ineffectivity in these arguments.

27 March, 2015 at 5:33 am

AnonymousWhich (hypothetical) bounds on are known to imply the twin prime conjecture?

27 March, 2015 at 6:05 am

Terence TaoRoughly speaking one needs bounds on for various that gain a couple of powers of over the trivial bound. (See Proposition 3 of https://terrytao.wordpress.com/2011/11/21/the-bourgain-sarnak-ziegler-orthogonality-criterion/ for some related computations.)

28 March, 2015 at 1:44 am

AnonymousIs it possible to improve the estimates in theorem 1 by assuming the general (!) tuple to be admissible (or have another “nice” property) ?

28 March, 2015 at 10:49 am

Terence TaoFor the method of proof, one needs as much averaging as possible, so restricting the range of possible $h_1,\dots,h_k$ would lead to inferior estimates. On the other hand, admissibility (or similar conditions) are not expected to help: the Mobius/Liouville randomness heuristic suggests that one should get square root decay uniformly for all choices of (assuming they are not too large compared with ).

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