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 have
In 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.)
29 comments
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17 March, 2015 at 11:39 pm
Anonymous
Is there any (probabilistic) prediction for the true size of the
term in theorem 1 ?
18 March, 2015 at 12:16 pm
Terence Tao
Standard probabilistic heuristics suggest that this expression should be of size
for any
.
18 March, 2015 at 5:50 am
Joel Moreira
This 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 Tao
Not 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
Anonymous
What 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 Moreira
I 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
Anonymous
Thank 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
brad
You’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
brad
This 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 Tao
Unfortunately 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
Anonymous
Is 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 Tao
Yes, 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
Anonymous
Are the implied constants in theorem 1.6 (in the arxiv paper) effective?
25 March, 2015 at 8:11 am
Terence Tao
Yes; 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
Anonymous
Which (hypothetical) bounds on
are known to imply the twin prime conjecture?
27 March, 2015 at 6:05 am
Terence Tao
Roughly 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
Anonymous
Is 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 Tao
For 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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