Let denote the Liouville function. The prime number theorem is equivalent to the estimate
as , that is to say that exhibits cancellation on large intervals such as . This result can be improved to give cancellation on shorter intervals. For instance, using the known zero density estimates for the Riemann zeta function, one can establish that
as if for some fixed ; I believe this result is due to Ramachandra (see also Exercise 21 of this previous blog post), and in fact one could obtain a better error term on the right-hand side that for instance gained an arbitrary power of . On the Riemann hypothesis (or the weaker density hypothesis), it was known that the could be lowered to .
Early this year, there was a major breakthrough by Matomaki and Radziwill, who (among other things) showed that the asymptotic (1) was in fact valid for any with that went to infinity as , thus yielding cancellation on extremely short intervals. This has many further applications; for instance, this estimate, or more precisely its extension to other “non-pretentious” bounded multiplicative functions, was a key ingredient in my recent solution of the Erdös discrepancy problem, as well as in obtaining logarithmically averaged cases of Chowla’s conjecture, such as
It is of interest to twist the above estimates by phases such as the linear phase . In 1937, Davenport showed that
from which one can see that this is another averaged form of Chowla’s conjecture (stronger than the one I was able to prove with Matomaki and Radziwill, but a consequence of the unaveraged Chowla conjecture). If one inserted such a bound into the machinery I used to solve the Erdös discrepancy problem, it should lead to further averaged cases of Chowla’s conjecture, such as
though I have not fully checked the details of this implication. It should also have a number of new implications for sign patterns of the Liouville function, though we have not explored these in detail yet.
One can write (4) equivalently in the form
uniformly for all -dependent phases . In contrast, (3) is equivalent to the subcase of (6) when the linear phase coefficient is independent of . This dependency of on seems to necessitate some highly nontrivial additive combinatorial analysis of the function in order to establish (4) when is small. To date, this analysis has proven to be elusive, but I would like to record what one can do with more classical methods like Vaughan’s identity, namely:
The values of in this range are far too large to yield implications such as new cases of the Chowla conjecture, but it appears that the exponent is the limit of “classical” methods (at least as far as I was able to apply them), in the sense that one does not do any combinatorial analysis on the function , nor does one use modern equidistribution results on “Type III sums” that require deep estimates on Kloosterman-type sums. The latter may shave a little bit off of the exponent, but I don’t see how one would ever hope to go below without doing some non-trivial combinatorics on the function . UPDATE: I have come across this paper of Zhan which uses mean-value theorems for L-functions to lower the exponent to .
Let me now sketch the proof of the proposition, omitting many of the technical details. We first remark that known estimates on sums of the Liouville function (or similar functions such as the von Mangoldt function) in short arithmetic progressions, based on zero-density estimates for Dirichlet -functions, can handle the “major arc” case of (4) (or (6)) where is restricted to be of the form for (the exponent here being of the same numerology as the exponent in the classical result of Ramachandra, tied to the best zero density estimates currently available); for instance a modification of the arguments in this recent paper of Koukoulopoulos would suffice. Thus we can restrict attention to “minor arc” values of (or , using the interpretation of (6)).
Next, one breaks up (or the closely related Möbius function) into Dirichlet convolutions using one of the standard identities (e.g. Vaughan’s identity or Heath-Brown’s identity), as discussed for instance in this previous post (which is focused more on the von Mangoldt function, but analogous identities exist for the Liouville and Möbius functions). The exact choice of identity is not terribly important, but the upshot is that can be decomposed into terms, each of which is either of the “Type I” form
for some coefficients that are roughly of logarithmic size on the average, and scales with and , or else of the “Type II” form
for some coefficients that are roughly of logarithmic size on the average, and scales with and . As discussed in the previous post, the exponent is a natural barrier in these identities if one is unwilling to also consider “Type III” type terms which are roughly of the shape of the third divisor function .
A Type I sum makes a contribution to that can be bounded (via Cauchy-Schwarz) in terms of an expression such as
The inner sum exhibits a lot of cancellation unless is within of an integer. (Here, “a lot” should be loosely interpreted as “gaining many powers of over the trivial bound”.) Since is significantly larger than , standard Vinogradov-type manipulations (see e.g. Lemma 13 of these previous notes) show that this bad case occurs for many only when is “major arc”, which is the case we have specifically excluded. This lets us dispose of the Type I contributions.
A Type II sum makes a contribution to roughly of the form
We can break this up into a number of sums roughly of the form
for ; note that the range is non-trivial because is much larger than . Applying the usual bilinear sum Cauchy-Schwarz methods (e.g. Theorem 14 of these notes) we conclude that there is a lot of cancellation unless one has for some . But with , is well below the threshold for the definition of major arc, so we can exclude this case and obtain the required cancellation.