Let ${\lambda}$ denote the Liouville function. The prime number theorem is equivalent to the estimate

$\displaystyle \sum_{n \leq x} \lambda(n) = o(x)$

as ${x \rightarrow \infty}$, that is to say that ${\lambda}$ exhibits cancellation on large intervals such as ${[1,x]}$. 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

$\displaystyle \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n)|\ dx = o( HX ) \ \ \ \ \ (1)$

as ${X \rightarrow \infty}$ if ${X^{1/6+\varepsilon} \leq H \leq X}$ for some fixed ${\varepsilon>0}$; 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 ${\log X}$. On the Riemann hypothesis (or the weaker density hypothesis), it was known that the ${X^{1/6+\varepsilon}}$ could be lowered to ${X^\varepsilon}$.

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 ${H = H(X)}$ with ${H \leq X}$ that went to infinity as ${X \rightarrow \infty}$, 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

$\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1)}{n} = o(\log x). \ \ \ \ \ (2)$

It is of interest to twist the above estimates by phases such as the linear phase ${n \mapsto e(\alpha n) := e^{2\pi i \alpha n}}$. In 1937, Davenport showed that

$\displaystyle \sup_\alpha |\sum_{n \leq x} \lambda(n) e(\alpha n)| \ll_A x \log^{-A} x$

which of course improves the prime number theorem. Recently with Matomaki and Radziwill, we obtained a common generalisation of this estimate with (1), showing that

$\displaystyle \sup_\alpha \int_X^{2X} |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(HX) \ \ \ \ \ (3)$

as ${X \rightarrow \infty}$, for any ${H = H(X) \leq X}$ that went to infinity as ${X \rightarrow \infty}$. We were able to use this estimate to obtain an averaged form of Chowla’s conjecture.

In that paper, we asked whether one could improve this estimate further by moving the supremum inside the integral, that is to say to establish the bound

$\displaystyle \int_X^{2X} \sup_\alpha |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o(HX) \ \ \ \ \ (4)$

as ${X \rightarrow \infty}$, for any ${H = H(X) \leq X}$ that went to infinity as ${X \rightarrow \infty}$. This bound is asserting that ${\lambda}$ is locally Fourier-uniform on most short intervals; it can be written equivalently in terms of the “local Gowers ${U^2}$ norm” as

$\displaystyle \int_X^{2X} \sum_{1 \leq a \leq H} |\sum_{x \leq n \leq x+H} \lambda(n) \lambda(n+a)|^2\ dx = o( H^3 X )$

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

$\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+1) \lambda(n+2)}{n} = o(\log x), \ \ \ \ \ (5)$

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

$\displaystyle \int_X^{2X} \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )\ dx = o(HX) \ \ \ \ \ (6)$

uniformly for all ${x}$-dependent phases ${\alpha(x), \beta(x)}$. In contrast, (3) is equivalent to the subcase of (6) when the linear phase coefficient ${\alpha(x)}$ is independent of ${x}$. This dependency of ${\alpha(x)}$ on ${x}$ seems to necessitate some highly nontrivial additive combinatorial analysis of the function ${x \mapsto \alpha(x)}$ in order to establish (4) when ${H}$ 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:

Proposition 1 The estimate (4) (or equivalently (6)) holds in the range ${X^{2/3+\varepsilon} \leq H \leq X}$ for any fixed ${\varepsilon>0}$. (In fact one can improve the right-hand side by an arbitrary power of ${\log X}$ in this case.)

The values of ${H}$ in this range are far too large to yield implications such as new cases of the Chowla conjecture, but it appears that the ${2/3}$ 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 ${x \mapsto \alpha(x)}$, 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 ${2/3}$ exponent, but I don’t see how one would ever hope to go below ${1/2}$ without doing some non-trivial combinatorics on the function ${x \mapsto \alpha(x)}$. UPDATE: I have come across this paper of Zhan which uses mean-value theorems for L-functions to lower the ${2/3}$ exponent to ${5/8}$.

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 ${L}$-functions, can handle the “major arc” case of (4) (or (6)) where ${\alpha}$ is restricted to be of the form ${\alpha = \frac{a}{q} + O( X^{-1/6-\varepsilon} )}$ for ${q = O(\log^{O(1)} X)}$ (the exponent here being of the same numerology as the ${X^{1/6+\varepsilon}}$ 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 ${\alpha}$ (or ${\alpha(x)}$, using the interpretation of (6)).

Next, one breaks up ${\lambda}$ (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 ${\lambda(n)}$ can be decomposed into ${\log^{O(1)} X}$ terms, each of which is either of the “Type I” form

$\displaystyle \sum_{d \sim D; m \sim M: dm=n} a_d$

for some coefficients ${a_d}$ that are roughly of logarithmic size on the average, and scales ${D, M}$ with ${D \ll X^{2/3}}$ and ${DM \sim X}$, or else of the “Type II” form

$\displaystyle \sum_{d \sim D; m \sim M: dm=n} a_d b_m$

for some coefficients ${a_d, b_m}$ that are roughly of logarithmic size on the average, and scales ${D,M}$ with ${X^{1/3} \ll D,M \ll X^{2/3}}$ and ${DM \sim X}$. As discussed in the previous post, the ${2/3}$ 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 ${\tau_3(n) := \sum_{d_1d_2d_3=1} 1}$.

A Type I sum makes a contribution to ${ \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )}$ that can be bounded (via Cauchy-Schwarz) in terms of an expression such as

$\displaystyle \sum_{d \sim D} | \sum_{x/d \leq m \leq x/d+H/d} e(\alpha(x) dm )|^2.$

The inner sum exhibits a lot of cancellation unless ${\alpha(x) d}$ is within ${O(D/H)}$ of an integer. (Here, “a lot” should be loosely interpreted as “gaining many powers of ${\log X}$ over the trivial bound”.) Since ${H}$ is significantly larger than ${D}$, standard Vinogradov-type manipulations (see e.g. Lemma 13 of these previous notes) show that this bad case occurs for many ${d}$ only when ${\alpha}$ 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 ${ \sum_{x \leq n \leq x+H} \lambda(n) e( \alpha(x) n + \beta(x) )}$ roughly of the form

$\displaystyle \sum_{d \sim D} | \sum_{x/d \leq m \leq x/d+H/d} b_m e(\alpha(x) dm)|.$

We can break this up into a number of sums roughly of the form

$\displaystyle \sum_{d = d_0 + O( H / M )} | \sum_{x/d_0 \leq m \leq x/d_0 + H/D} b_m e(\alpha(x) dm)|$

for ${d_0 \sim D}$; note that the ${d}$ range is non-trivial because ${H}$ is much larger than ${M}$. 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 ${\alpha(x) = a/q + O( \frac{X \log^{O(1)} X}{H^2} )}$ for some ${q = O(\log^{O(1)} X)}$. But with ${H \geq X^{2/3+\varepsilon}}$, ${X \log^{O(1)} X/H^2}$ is well below the threshold ${X^{-1/6-\varepsilon}}$ for the definition of major arc, so we can exclude this case and obtain the required cancellation.