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In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number of useful topologies on them, including the operator norm topology, strong operator topology, and the weak operator topology. For function spaces, one can use topologies associated to various modes of convergence, such as uniform convergence, pointwise convergence, locally uniform convergence, or convergence in the sense of distributions. (A small minority of such modes are not topologisable, though, the most common of which is pointwise almost everywhere convergence; see Exercise 8 of this previous post).

Some of these topologies are much stronger than others (in that they contain many more open sets, or equivalently that they have many fewer convergent sequences and nets). However, even the weakest topologies used in analysis (e.g. convergence in distributions) tend to be Hausdorff, since this at least ensures the uniqueness of limits of sequences and nets, which is a fundamentally useful feature for analysis. On the other hand, some Hausdorff topologies used are “better” than others in that many more analysis tools are available for those topologies. In particular, topologies that come from Banach space norms are particularly valued, as such topologies (and their attendant norm and metric structures) grant access to many convenient additional results such as the Baire category theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.

Of course, most topologies placed on a vector space will not come from Banach space norms. For instance, if one takes the space {C_0({\bf R})} of continuous functions on {{\bf R}} that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm {\| \|_{L^\infty}}), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. {L^1} convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.

I recently realised (while teaching a graduate class in real analysis) that the closed graph theorem provides a quick explanation for why Banach space topologies are so rare:

Proposition 1 Let {V = (V, {\mathcal F})} be a Hausdorff topological vector space. Then, up to equivalence of norms, there is at most one norm {\| \|} one can place on {V} so that {(V,\| \|)} is a Banach space whose topology is at least as strong as {{\mathcal F}}. In particular, there is at most one topology stronger than {{\mathcal F}} that comes from a Banach space norm.

Proof: Suppose one had two norms {\| \|_1, \| \|_2} on {V} such that {(V, \| \|_1)} and {(V, \| \|_2)} were both Banach spaces with topologies stronger than {{\mathcal F}}. Now consider the graph of the identity function {\hbox{id}: V \rightarrow V} from the Banach space {(V, \| \|_1)} to the Banach space {(V, \| \|_2)}. This graph is closed; indeed, if {(x_n,x_n)} is a sequence in this graph that converged in the product topology to {(x,y)}, then {x_n} converges to {x} in {\| \|_1} norm and hence in {{\mathcal F}}, and similarly {x_n} converges to {y} in {\| \|_2} norm and hence in {{\mathcal F}}. But limits are unique in the Hausdorff topology {{\mathcal F}}, so {x=y}. Applying the closed graph theorem (see also previous discussions on this theorem), we see that the identity map is continuous from {(V, \| \|_1)} to {(V, \| \|_2)}; similarly for the inverse. Thus the norms {\| \|_1, \| \|_2} are equivalent as claimed. \Box

By using various generalisations of the closed graph theorem, one can generalise the above proposition to Fréchet spaces, or even to F-spaces. The proposition can fail if one drops the requirement that the norms be stronger than a specified Hausdorff topology; indeed, if {V} is infinite dimensional, one can use a Hamel basis of {V} to construct a linear bijection on {V} that is unbounded with respect to a given Banach space norm {\| \|}, and which can then be used to give an inequivalent Banach space structure on {V}.

One can interpret Proposition 1 as follows: once one equips a vector space with some “weak” (but still Hausdorff) topology, there is a canonical choice of “strong” topology one can place on that space that is stronger than the “weak” topology but arises from a Banach space structure (or at least a Fréchet or F-space structure), provided that at least one such structure exists. In the case of function spaces, one can usually use the topology of convergence in distribution as the “weak” Hausdorff topology for this purpose, since this topology is weaker than almost all of the other topologies used in analysis. This helps justify the common practice of describing a Banach or Fréchet function space just by giving the set of functions that belong to that space (e.g. {{\mathcal S}({\bf R}^n)} is the space of Schwartz functions on {{\bf R}^n}) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on {{\mathcal S}({\bf R}^n)} given by the usual Schwartz space seminorms).

Of course, there are still some topological vector spaces which have no “strong topology” arising from a Banach space at all. Consider for instance the space {c_c({\bf N})} of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm {\| \|} stronger than this topology that makes this space a Banach space. For, if there were, then letting {e_1,e_2,e_3,\dots} be the standard basis of {c_c({\bf N})}, the series {\sum_{n=1}^\infty 2^{-n} e_n / \| e_n \|} would have to converge in {\| \|}, and hence pointwise, to an element of {c_c({\bf N})}, but the only available pointwise limit for this series lies outside of {c_c({\bf N})}. But I do not know if there is an easily checkable criterion to test whether a given vector space (equipped with a Hausdorff “weak” toplogy) can be equipped with a stronger Banach space (or Fréchet space or {F}-space) topology.

There is a very nice recent paper by Lemke Oliver and Soundararajan (complete with a popular science article about it by the consistently excellent Erica Klarreich for Quanta) about a surprising (but now satisfactorily explained) bias in the distribution of pairs of consecutive primes {p_n, p_{n+1}} when reduced to a small modulus {q}.

This phenomenon is superficially similar to the more well known Chebyshev bias concerning the reduction of a single prime {p_n} to a small modulus {q}, but is in fact a rather different (and much stronger) bias than the Chebyshev bias, and seems to arise from a completely different source. The Chebyshev bias asserts, roughly speaking, that a randomly selected prime {p} of a large magnitude {x} will typically (though not always) be slightly more likely to be a quadratic non-residue modulo {q} than a quadratic residue, but the bias is small (the difference in probabilities is only about {O(1/\sqrt{x})} for typical choices of {x}), and certainly consistent with known or conjectured positive results such as Dirichlet’s theorem or the generalised Riemann hypothesis. The reason for the Chebyshev bias can be traced back to the von Mangoldt explicit formula which relates the distribution of the von Mangoldt function {\Lambda} modulo {q} with the zeroes of the {L}-functions with period {q}. This formula predicts (assuming some standard conjectures like GRH) that the von Mangoldt function {\Lambda} is quite unbiased modulo {q}. The von Mangoldt function is mostly concentrated in the primes, but it also has a medium-sized contribution coming from squares of primes, which are of course all located in the quadratic residues modulo {q}. (Cubes and higher powers of primes also make a small contribution, but these are quite negligible asymptotically.) To balance everything out, the contribution of the primes must then exhibit a small preference towards quadratic non-residues, and this is the Chebyshev bias. (See this article of Rubinstein and Sarnak for a more technical discussion of the Chebyshev bias, and this survey of Granville and Martin for an accessible introduction. The story of the Chebyshev bias is also related to Skewes’ number, once considered the largest explicit constant to naturally appear in a mathematical argument.)

The paper of Lemke Oliver and Soundararajan considers instead the distribution of the pairs {(p_n \hbox{ mod } q, p_{n+1} \hbox{ mod } q)} for small {q} and for large consecutive primes {p_n, p_{n+1}}, say drawn at random from the primes comparable to some large {x}. For sake of discussion let us just take {q=3}. Then all primes {p_n} larger than {3} are either {1 \hbox{ mod } 3} or {2 \hbox{ mod } 3}; Chebyshev’s bias gives a very slight preference to the latter (of order {O(1/\sqrt{x})}, as discussed above), but apart from this, we expect the primes to be more or less equally distributed in both classes. For instance, assuming GRH, the probability that {p_n} lands in {1 \hbox{ mod } 3} would be {1/2 + O( x^{-1/2+o(1)} )}, and similarly for {2 \hbox{ mod } 3}.

In view of this, one would expect that up to errors of {O(x^{-1/2+o(1)})} or so, the pair {(p_n \hbox{ mod } 3, p_{n+1} \hbox{ mod } 3)} should be equally distributed amongst the four options {(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}, {(1 \hbox{ mod } 3, 2 \hbox{ mod } 3)}, {(2 \hbox{ mod } 3, 1 \hbox{ mod } 3)}, {(2 \hbox{ mod } 3, 2 \hbox{ mod } 3)}, thus for instance the probability that this pair is {(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)} would naively be expected to be {1/4 + O(x^{-1/2+o(1)})}, and similarly for the other three tuples. These assertions are not yet proven (although some non-trivial upper and lower bounds for such probabilities can be obtained from recent work of Maynard).

However, Lemke Oliver and Soundararajan argue (backed by both plausible heuristic arguments (based ultimately on the Hardy-Littlewood prime tuples conjecture), as well as substantial numerical evidence) that there is a significant bias away from the tuples {(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)} and {(2 \hbox{ mod } 3, 2 \hbox{ mod } 3)} – informally, adjacent primes don’t like being in the same residue class! For instance, they predict that the probability of attaining {(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)} is in fact

\displaystyle  \frac{1}{4} - \frac{1}{8} \frac{\log\log x}{\log x} + O( \frac{1}{\log x} )

with similar predictions for the other three pairs (in fact they give a somewhat more precise prediction than this). The magnitude of this bias, being comparable to {\log\log x / \log x}, is significantly stronger than the Chebyshev bias of {O(1/\sqrt{x})}.

One consequence of this prediction is that the prime gaps {p_{n+1}-p_n} are slightly less likely to be divisible by {3} than naive random models of the primes would predict. Indeed, if the four options {(1 \hbox{ mod } 3, 1 \hbox{ mod } 3)}, {(1 \hbox{ mod } 3, 2 \hbox{ mod } 3)}, {(2 \hbox{ mod } 3, 1 \hbox{ mod } 3)}, {(2 \hbox{ mod } 3, 2 \hbox{ mod } 3)} all occurred with equal probability {1/4}, then {p_{n+1}-p_n} should equal {0 \hbox{ mod } 3} with probability {1/2}, and {1 \hbox{ mod } 3} and {2 \hbox{ mod } 3} with probability {1/4} each (as would be the case when taking the difference of two random numbers drawn from those integers not divisible by {3}); but the Lemke Oliver-Soundararajan bias predicts that the probability of {p_{n+1}-p_n} being divisible by three should be slightly lower, being approximately {1/2 - \frac{1}{4} \frac{\log\log x}{\log x}}.

Below the fold we will give a somewhat informal justification of (a simplified version of) this phenomenon, based on the Lemke Oliver-Soundararajan calculation using the prime tuples conjecture.

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I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup. (I recently spoke on this work in the recent conference in Princeton in honour of Sergiu Klainerman; my slides for that talk are here.)

One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity {\omega = \nabla \times u}, in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become

\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u + \nu \Delta \omega

\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).

(Throughout this post we will be working in three spatial dimensions {{\bf R}^3}.) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.

Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case {\nu=0} of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity (which, as explained in the above slides, heuristically means that the effects of viscosity are negligible when compared against the nonlinear components of the equation). In vorticity-stream formulation, the Euler equations can be written as

\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u

\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).

As discussed in this previous blog post, a natural generalisation of this system of equations is the system

\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (1)

 

\displaystyle u = T (-\Delta)^{-1} (\nabla \times \omega).

where {T} is a linear operator on divergence-free vector fields that is “zeroth order” in some sense; ideally it should also be invertible, self-adjoint, and positive definite (in order to have a Hamiltonian that is comparable to the kinetic energy {\frac{1}{2} \int_{{\bf R}^3} |u|^2}). (In the previous blog post, it was observed that the surface quasi-geostrophic (SQG) equation could be embedded in a system of the form (1).) The system (1) has many features in common with the Euler equations; for instance vortex lines are transported by the velocity field {u}, and Kelvin’s circulation theorem is still valid.

So far, I have not been able to fully achieve this goal. However, I have the following partial result, stated somewhat informally:

Theorem 1 There is a “zeroth order” linear operator {T} (which, unfortunately, is not invertible, self-adjoint, or positive definite) for which the system (1) exhibits smooth solutions that blowup in finite time.

The operator {T} constructed is not quite a zeroth-order pseudodifferential operator; it is instead merely in the “forbidden” symbol class {S^0_{1,1}}, and more precisely it takes the form

\displaystyle T v = \sum_{j \in {\bf Z}} 2^{3j} \langle v, \phi_j \rangle \psi_j \ \ \ \ \ (2)

 

for some compactly supported divergence-free {\phi,\psi} of mean zero with

\displaystyle \phi_j(x) := \phi(2^j x); \quad \psi_j(x) := \psi(2^j x)

being {L^2} rescalings of {\phi,\psi}. This operator is still bounded on all {L^p({\bf R}^3)} spaces {1 < p < \infty}, and so is arguably still a zeroth order operator, though not as convincingly as I would like. Another, less significant, issue with the result is that the solution constructed does not have good spatial decay properties, but this is mostly for convenience and it is likely that the construction can be localised to give solutions that have reasonable decay in space. But the biggest drawback of this theorem is the fact that {T} is not invertible, self-adjoint, or positive definite, so in particular there is no non-negative Hamiltonian for this equation. It may be that some modification of the arguments below can fix these issues, but I have so far been unable to do so. Still, the construction does show that the circulation theorem is insufficient by itself to prevent blowup.

We sketch the proof of the above theorem as follows. We use the barrier method, introducing the time-varying hyperboloid domains

\displaystyle \Omega(t) := \{ (r,\theta,z): r^2 \leq 1-t + z^2 \}

for {t>0} (expressed in cylindrical coordinates {(r,\theta,z)}). We will select initial data {\omega(0)} to be {\omega(0,r,\theta,z) = (0,0,\eta(r))} for some non-negative even bump function {\eta} supported on {[-1,1]}, normalised so that

\displaystyle \int\int \eta(r)\ r dr d\theta = 1;

in particular {\omega(0)} is divergence-free supported in {\Omega(0)}, with vortex lines connecting {z=-\infty} to {z=+\infty}. Suppose for contradiction that we have a smooth solution {\omega} to (1) with this initial data; to simplify the discussion we assume that the solution behaves well at spatial infinity (this can be justified with the choice (2) of vorticity-stream operator, but we will not do so here). Since the domains {\Omega(t)} disconnect {z=-\infty} from {z=+\infty} at time {t=1}, there must exist a time {0 < T_* < 1} which is the first time where the support of {\omega(T_*)} touches the boundary of {\Omega(T_*)}, with {\omega(t)} supported in {\Omega(t)}.

From (1) we see that the support of {\omega(t)} is transported by the velocity field {u(t)}. Thus, at the point of contact of the support of {\omega(T_*)} with the boundary of {\Omega(T_*)}, the inward component of the velocity field {u(T_*)} cannot exceed the inward velocity of {\Omega(T_*)}. We will construct the functions {\phi,\psi} so that this is not the case, leading to the desired contradiction. (Geometrically, what is going on here is that the operator {T} is pinching the flow to pass through the narrow cylinder {\{ z, r = O( \sqrt{1-t} )\}}, leading to a singularity by time {t=1} at the latest.)

First we observe from conservation of circulation, and from the fact that {\omega(t)} is supported in {\Omega(t)}, that the integrals

\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta

are constant in both space and time for {0 \leq t \leq T_*}. From the choice of initial data we thus have

\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta = 1

for all {t \leq T_*} and all {z}. On the other hand, if {T} is of the form (2) with {\phi = \nabla \times \eta} for some bump function {\eta = (0,0,\eta_z)} that only has {z}-components, then {\phi} is divergence-free with mean zero, and

\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = 2^{-j} \langle (-\Delta) (\nabla \times \omega), \nabla \times \eta_j \rangle

\displaystyle = 2^{-j} \langle \omega, \eta_j \rangle

\displaystyle = 2^{-j} \int\int\int \omega_z(t,r,\theta,z) \eta_z(2^j r, \theta, 2^j z)\ r dr d\theta dz,

where {\eta_j(x) := \eta(2^j x)}. We choose {\eta_z} to be supported in the slab {\{ C \leq z \leq 2C\}} for some large constant {C}, and to equal a function {f(z)} depending only on {z} on the cylinder {\{ C \leq z \leq 2C; r \leq 10C \}}, normalised so that {\int f(z)\ dz = 1}. If {C/2^j \geq (1-t)^{1/2}}, then {\Omega(t)} passes through this cylinder, and we conclude that

\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = -2^{-j} \int f(2^j z)\ dz

\displaystyle = 2^{-2j}.

Inserting ths into (2), (1) we conclude that

\displaystyle u = \sum_{j: C/2^j \geq (1-t)^{1/2}} 2^j \psi_j + \sum_{j: C/2^j < (1-t)^{1/2}} c_j(t) \psi_j

for some coefficients {c_j(t)}. We will not be able to control these coefficients {c_j(t)}, but fortunately we only need to understand {u} on the boundary {\partial \Omega(t)}, for which {r+|z| \gg (1-t)^{1/2}}. So, if {\psi} happens to be supported on an annulus {1 \ll r+|z| \ll 1}, then {\psi_j} vanishes on {\partial \Omega(t)} if {C} is large enough. We then have

\displaystyle u = \sum_j 2^j \psi_j

on the boundary of {\partial \Omega(t)}.

Let {\Phi(r,\theta,z)} be a function of the form

\displaystyle \Phi(r,\theta,z) = C z \varphi(z/r)

where {\varphi} is a bump function supported on {[-2,2]} that equals {1} on {[-1,1]}. We can perform a dyadic decomposition {\Phi = \sum_j \Psi_j} where

\displaystyle \Psi_j(r,\theta,z) = \Phi(r,\theta,z) a(2^j r)

where {a} is a bump function supported on {[1/2,2]} with {\sum_j a(2^j r) = 1}. If we then set

\displaystyle \psi_j = \frac{2^{-j}}{r} (-\partial_z \Psi_j, 0, \partial_r \Psi_j)

then one can check that {\psi_j(x) = \psi(2^j x)} for a function {\psi} that is divergence-free and mean zero, and supported on the annulus {1 \ll r+|z| \ll 1}, and

\displaystyle \sum_j 2^j \psi_j = \frac{1}{r} (-\partial_z \Phi, 0, \partial_r \Phi)

so on {\partial \Omega(t)} (where {|z| \leq r}) we have

\displaystyle u = (-\frac{C}{r}, 0, 0 ).

One can manually check that the inward velocity of this vector on {\partial \Omega(t)} exceeds the inward velocity of {\Omega(t)} if {C} is large enough, and the claim follows.

Remark 2 The type of blowup suggested by this construction, where a unit amount of circulation is squeezed into a narrow cylinder, is of “Type II” with respect to the Navier-Stokes scaling, because Navier-Stokes-critical norms such {L^3({\bf R}^3)} (or at least {L^{3,\infty}({\bf R}^3)}) look like they stay bounded during this squeezing procedure (the velocity field is of size about {2^j} in cylinders of radius and length about {2^j}). So even if the various issues with {T} are repaired, it does not seem likely that this construction can be directly adapted to obtain a corresponding blowup for a Navier-Stokes type equation. To get a “Type I” blowup that is consistent with Kelvin’s circulation theorem, it seems that one needs to coil the vortex lines around a loop multiple times in order to get increased circulation in a small space. This seems possible to pull off to me – there don’t appear to be any unavoidable obstructions coming from topology, scaling, or conservation laws – but would require a more complicated construction than the one given above.

In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely

Theorem 1 (Two-dimensional Vinogradov main conjecture) One has

\displaystyle  \int_{[0,1]^2} |\sum_{j=0}^N e( j x + j^2 y)|^6\ dx dy \ll N^{3+o(1)}

as {N \rightarrow \infty}.

This particular case of the main conjecture has a classical proof using some elementary number theory. Indeed, the left-hand side can be viewed as the number of solutions to the system of equations

\displaystyle  j_1 + j_2 + j_3 = k_1 + k_2 + k_3

\displaystyle  j_1^2 + j_2^2 + j_3^2 = k_1^2 + k_2^2 + k_3^2

with {j_1,j_2,j_3,k_1,k_2,k_3 \in \{0,\dots,N\}}. These two equations can combine (using the algebraic identity {(a+b-c)^2 - (a^2+b^2-c^2) = 2 (a-c)(b-c)} applied to {(a,b,c) = (j_1,j_2,k_3), (k_1,k_2,j_3)}) to imply the further equation

\displaystyle  (j_1 - k_3) (j_2 - k_3) = (k_1 - j_3) (k_2 - j_3)

which, when combined with the divisor bound, shows that each {k_1,k_2,j_3} is associated to {O(N^{o(1)})} choices of {j_1,j_2,k_3} excluding diagonal cases when two of the {j_1,j_2,j_3,k_1,k_2,k_3} collide, and this easily yields Theorem 1. However, the Bourgain-Demeter-Guth argument (which, in the two dimensional case, is essentially contained in a previous paper of Bourgain and Demeter) does not require the divisor bound, and extends for instance to the the more general case where {j} ranges in a {1}-separated set of reals between {0} to {N}.

In this special case, the Bourgain-Demeter argument simplifies, as the lower dimensional inductive hypothesis becomes a simple {L^2} almost orthogonality claim, and the multilinear Kakeya estimate needed is also easy (collapsing to just Fubini’s theorem). Also one can work entirely in the context of the Vinogradov main conjecture, and not turn to the increased generality of decoupling inequalities (though this additional generality is convenient in higher dimensions). As such, I am presenting this special case as an introduction to the Bourgain-Demeter-Guth machinery.

We now give the specialisation of the Bourgain-Demeter argument to Theorem 1. It will suffice to establish the bound

\displaystyle  \int_{[0,1]^2} |\sum_{j=0}^N e( j x + j^2 y)|^p\ dx dy \ll N^{p/2+o(1)}

for all {4<p<6}, (where we keep {p} fixed and send {N} to infinity), as the {L^6} bound then follows by combining the above bound with the trivial bound {|\sum_{j=0}^N e( j x + j^2 x^2)| \ll N}. Accordingly, for any {\eta > 0} and {4<p<6}, we let {P(p,\eta)} denote the claim that

\displaystyle  \int_{[0,1]^2} |\sum_{j=0}^N e( j x + j^2 y)|^p\ dx dy \ll N^{p/2+\eta+o(1)}

as {N \rightarrow \infty}. Clearly, for any fixed {p}, {P(p,\eta)} holds for some large {\eta}, and it will suffice to establish

Proposition 2 Let {4<p<6}, and let {\eta>0} be such that {P(p,\eta)} holds. Then there exists {0 < \eta' < \eta} such that {P(p,\eta')} holds.

Indeed, this proposition shows that for {4<p<6}, the infimum of the {\eta} for which {P(p,\eta)} holds is zero.

We prove the proposition below the fold, using a simplified form of the methods discussed in the previous blog post. To simplify the exposition we will be a bit cavalier with the uncertainty principle, for instance by essentially ignoring the tails of rapidly decreasing functions.

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Given any finite collection of elements {(f_i)_{i \in I}} in some Banach space {X}, the triangle inequality tells us that

\displaystyle \| \sum_{i \in I} f_i \|_X \leq \sum_{i \in I} \|f_i\|_X.

However, when the {f_i} all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. For instance, if {X} is a Hilbert space and the {f_i} are mutually orthogonal, we have the Pythagorean theorem

\displaystyle \| \sum_{i \in I} f_i \|_X = (\sum_{i \in I} \|f_i\|_X^2)^{1/2}.

For sake of comparison, from the triangle inequality and Cauchy-Schwarz one has the general inequality

\displaystyle \| \sum_{i \in I} f_i \|_X \leq (\# I)^{1/2} (\sum_{i \in I} \|f_i\|_X^2)^{1/2} \ \ \ \ \ (1)

 

for any finite collection {(f_i)_{i \in I}} in any Banach space {X}, where {\# I} denotes the cardinality of {I}. Thus orthogonality in a Hilbert space yields “square root cancellation”, saving a factor of {(\# I)^{1/2}} or so over the trivial bound coming from the triangle inequality.

More generally, let us somewhat informally say that a collection {(f_i)_{i \in I}} exhibits decoupling in {X} if one has the Pythagorean-like inequality

\displaystyle \| \sum_{i \in I} f_i \|_X \ll_\varepsilon (\# I)^\varepsilon (\sum_{i \in I} \|f_i\|_X^2)^{1/2}

for any {\varepsilon>0}, thus one obtains almost the full square root cancellation in the {X} norm. The theory of almost orthogonality can then be viewed as the theory of decoupling in Hilbert spaces such as {L^2({\bf R}^n)}. In {L^p} spaces for {p < 2} one usually does not expect this sort of decoupling; for instance, if the {f_i} are disjointly supported one has

\displaystyle \| \sum_{i \in I} f_i \|_{L^p} = (\sum_{i \in I} \|f_i\|_{L^p}^p)^{1/p}

and the right-hand side can be much larger than {(\sum_{i \in I} \|f_i\|_{L^p}^2)^{1/2}} when {p < 2}. At the opposite extreme, one usually does not expect to get decoupling in {L^\infty}, since one could conceivably align the {f_i} to all attain a maximum magnitude at the same location with the same phase, at which point the triangle inequality in {L^\infty} becomes sharp.

However, in some cases one can get decoupling for certain {2 < p < \infty}. For instance, suppose we are in {L^4}, and that {f_1,\dots,f_N} are bi-orthogonal in the sense that the products {f_i f_j} for {1 \leq i < j \leq N} are pairwise orthogonal in {L^2}. Then we have

\displaystyle \| \sum_{i = 1}^N f_i \|_{L^4}^2 = \| (\sum_{i=1}^N f_i)^2 \|_{L^2}

\displaystyle = \| \sum_{1 \leq i,j \leq N} f_i f_j \|_{L^2}

\displaystyle \ll (\sum_{1 \leq i,j \leq N} \|f_i f_j \|_{L^2}^2)^{1/2}

\displaystyle = \| (\sum_{1 \leq i,j \leq N} |f_i f_j|^2)^{1/2} \|_{L^2}

\displaystyle = \| \sum_{i=1}^N |f_i|^2 \|_{L^2}

\displaystyle \leq \sum_{i=1}^N \| |f_i|^2 \|_{L^2}

\displaystyle = \sum_{i=1}^N \|f_i\|_{L^4}^2

giving decoupling in {L^4}. (Similarly if each of the {f_i f_j} is orthogonal to all but {O_\varepsilon( N^\varepsilon )} of the other {f_{i'} f_{j'}}.) A similar argument also gives {L^6} decoupling when one has tri-orthogonality (with the {f_i f_j f_k} mostly orthogonal to each other), and so forth. As a slight variant, Khintchine’s inequality also indicates that decoupling should occur for any fixed {2 < p < \infty} if one multiplies each of the {f_i} by an independent random sign {\epsilon_i \in \{-1,+1\}}.

In recent years, Bourgain and Demeter have been establishing decoupling theorems in {L^p({\bf R}^n)} spaces for various key exponents of {2 < p < \infty}, in the “restriction theory” setting in which the {f_i} are Fourier transforms of measures supported on different portions of a given surface or curve; this builds upon the earlier decoupling theorems of Wolff. In a recent paper with Guth, they established the following decoupling theorem for the curve {\gamma({\bf R}) \subset {\bf R}^n} parameterised by the polynomial curve

\displaystyle \gamma: t \mapsto (t, t^2, \dots, t^n).

For any ball {B = B(x_0,r)} in {{\bf R}^n}, let {w_B: {\bf R}^n \rightarrow {\bf R}^+} denote the weight

\displaystyle w_B(x) := \frac{1}{(1 + \frac{|x-x_0|}{r})^{100n}},

which should be viewed as a smoothed out version of the indicator function {1_B} of {B}. In particular, the space {L^p(w_B) = L^p({\bf R}^n, w_B(x)\ dx)} can be viewed as a smoothed out version of the space {L^p(B)}. For future reference we observe a fundamental self-similarity of the curve {\gamma({\bf R})}: any arc {\gamma(I)} in this curve, with {I} a compact interval, is affinely equivalent to the standard arc {\gamma([0,1])}.

Theorem 1 (Decoupling theorem) Let {n \geq 1}. Subdivide the unit interval {[0,1]} into {N} equal subintervals {I_i} of length {1/N}, and for each such {I_i}, let {f_i: {\bf R}^n \rightarrow {\bf R}} be the Fourier transform

\displaystyle f_i(x) = \int_{\gamma(I_i)} e(x \cdot \xi)\ d\mu_i(\xi)

of a finite Borel measure {\mu_i} on the arc {\gamma(I_i)}, where {e(\theta) := e^{2\pi i \theta}}. Then the {f_i} exhibit decoupling in {L^{n(n+1)}(w_B)} for any ball {B} of radius {N^n}.

Orthogonality gives the {n=1} case of this theorem. The bi-orthogonality type arguments sketched earlier only give decoupling in {L^p} up to the range {2 \leq p \leq 2n}; the point here is that we can now get a much larger value of {n}. The {n=2} case of this theorem was previously established by Bourgain and Demeter (who obtained in fact an analogous theorem for any curved hypersurface). The exponent {n(n+1)} (and the radius {N^n}) is best possible, as can be seen by the following basic example. If

\displaystyle f_i(x) := \int_{I_i} e(x \cdot \gamma(\xi)) g_i(\xi)\ d\xi

where {g_i} is a bump function adapted to {I_i}, then standard Fourier-analytic computations show that {f_i} will be comparable to {1/N} on a rectangular box of dimensions {N \times N^2 \times \dots \times N^n} (and thus volume {N^{n(n+1)/2}}) centred at the origin, and exhibit decay away from this box, with {\|f_i\|_{L^{n(n+1)}(w_B)}} comparable to

\displaystyle 1/N \times (N^{n(n+1)/2})^{1/(n(n+1))} = 1/\sqrt{N}.

On the other hand, {\sum_{i=1}^N f_i} is comparable to {1} on a ball of radius comparable to {1} centred at the origin, so {\|\sum_{i=1}^N f_i\|_{L^{n(n+1)}(w_B)}} is {\gg 1}, which is just barely consistent with decoupling. This calculation shows that decoupling will fail if {n(n+1)} is replaced by any larger exponent, and also if the radius of the ball {B} is reduced to be significantly smaller than {N^n}.

This theorem has the following consequence of importance in analytic number theory:

Corollary 2 (Vinogradov main conjecture) Let {s, n, N \geq 1} be integers, and let {\varepsilon > 0}. Then

\displaystyle \int_{[0,1]^n} |\sum_{j=1}^N e( j x_1 + j^2 x_2 + \dots + j^n x_n)|^{2s}\ dx_1 \dots dx_n

\displaystyle \ll_{\varepsilon,s,n} N^{s+\varepsilon} + N^{2s - \frac{n(n+1)}{2}+\varepsilon}.

Proof: By the Hölder inequality (and the trivial bound of {N} for the exponential sum), it suffices to treat the critical case {s = n(n+1)/2}, that is to say to show that

\displaystyle \int_{[0,1]^n} |\sum_{j=1}^N e( j x_1 + j^2 x_2 + \dots + j^n x_n)|^{n(n+1)}\ dx_1 \dots dx_n \ll_{\varepsilon,n} N^{\frac{n(n+1)}{2}+\varepsilon}.

We can rescale this as

\displaystyle \int_{[0,N] \times [0,N^2] \times \dots \times [0,N^n]} |\sum_{j=1}^N e( x \cdot \gamma(j/N) )|^{n(n+1)}\ dx \ll_{\varepsilon,n} N^{3\frac{n(n+1)}{2}+\varepsilon}.

As the integrand is periodic along the lattice {N{\bf Z} \times N^2 {\bf Z} \times \dots \times N^n {\bf Z}}, this is equivalent to

\displaystyle \int_{[0,N^n]^n} |\sum_{j=1}^N e( x \cdot \gamma(j/N) )|^{n(n+1)}\ dx \ll_{\varepsilon,n} N^{\frac{n(n+1)}{2}+n^2+\varepsilon}.

The left-hand side may be bounded by {\ll \| \sum_{j=1}^N f_j \|_{L^{n(n+1)}(w_B)}^{n(n+1)}}, where {B := B(0,N^n)} and {f_j(x) := e(x \cdot \gamma(j/N))}. Since

\displaystyle \| f_j \|_{L^{n(n+1)}(w_B)} \ll (N^{n^2})^{\frac{1}{n(n+1)}},

the claim now follows from the decoupling theorem and a brief calculation. \Box

Using the Plancherel formula, one may equivalently (when {s} is an integer) write the Vinogradov main conjecture in terms of solutions {j_1,\dots,j_s,k_1,\dots,k_s \in \{1,\dots,N\}} to the system of equations

\displaystyle j_1^i + \dots + j_s^i = k_1^i + \dots + k_s^i \forall i=1,\dots,n,

but we will not use this formulation here.

A history of the Vinogradov main conjecture may be found in this survey of Wooley; prior to the Bourgain-Demeter-Guth theorem, the conjecture was solved completely for {n \leq 3}, or for {n > 3} and {s} either below {n(n+1)/2 - n/3 + O(n^{2/3})} or above {n(n-1)}, with the bulk of recent progress coming from the efficient congruencing technique of Wooley. It has numerous applications to exponential sums, Waring’s problem, and the zeta function; to give just one application, the main conjecture implies the predicted asymptotic for the number of ways to express a large number as the sum of {23} fifth powers (the previous best result required {28} fifth powers). The Bourgain-Demeter-Guth approach to the Vinogradov main conjecture, based on decoupling, is ostensibly very different from the efficient congruencing technique, which relies heavily on the arithmetic structure of the program, but it appears (as I have been told from second-hand sources) that the two methods are actually closely related, with the former being a sort of “Archimedean” version of the latter (with the intervals {I_i} in the decoupling theorem being analogous to congruence classes in the efficient congruencing method); hopefully there will be some future work making this connection more precise. One advantage of the decoupling approach is that it generalises to non-arithmetic settings in which the set {\{1,\dots,N\}} that {j} is drawn from is replaced by some other similarly separated set of real numbers. (A random thought – could this allow the Vinogradov-Korobov bounds on the zeta function to extend to Beurling zeta functions?)

Below the fold we sketch the Bourgain-Demeter-Guth argument proving Theorem 1.

I thank Jean Bourgain and Andrew Granville for helpful discussions.

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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.

A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum).

Theorem 1 (Halasz inequality) Let {f: {\bf N} \rightarrow {\bf C}} be a multiplicative function bounded in magnitude by {1}, and suppose that {x \geq 3}, {T \geq 1}, and { M \geq 0} are such that

\displaystyle \sum_{p \leq x} \frac{1 - \hbox{Re}(f(p) p^{-it})}{p} \geq M \ \ \ \ \ (1)

 

for all real numbers {t} with {|t| \leq T}. Then

\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{\sqrt{T}}.

As a qualitative corollary, we conclude (by standard compactness arguments) that if

\displaystyle \sum_{p} \frac{1 - \hbox{Re}(f(p) p^{-it})}{p} = +\infty

for all real {t}, then

\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) = o(1) \ \ \ \ \ (2)

 

as {x \rightarrow \infty}. In the more recent work of this paper of Granville and Soundararajan, the sharper bound

\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{T} + \frac{\log\log x}{\log x}

is obtained (with a more precise description of the {(1+M) e^{-M}} term).

The usual proofs of Halasz’s theorem are somewhat lengthy (though there has been a recent simplification, in forthcoming work of Granville, Harper, and Soundarajan). Below the fold I would like to give a relatively short proof of the following “cheap” version of the inequality, which has slightly weaker quantitative bounds, but still suffices to give qualitative conclusions such as (2).

Theorem 2 (Cheap Halasz inequality) Let {f: {\bf N} \rightarrow {\bf C}} be a multiplicative function bounded in magnitude by {1}. Let {T \geq 1} and {M \geq 0}, and suppose that {x} is sufficiently large depending on {T,M}. If (1) holds for all {|t| \leq T}, then

\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M/2} + \frac{1}{T}.

The non-optimal exponent {1/2} can probably be improved a bit by being more careful with the exponents, but I did not try to optimise it here. A similar bound appears in the first paper of Halasz on this topic.

The idea of the argument is to split {f} as a Dirichlet convolution {f = f_1 * f_2 * f_3} where {f_1,f_2,f_3} is the portion of {f} coming from “small”, “medium”, and “large” primes respectively (with the dividing line between the three types of primes being given by various powers of {x}). Using a Perron-type formula, one can express this convolution in terms of the product of the Dirichlet series of {f_1,f_2,f_3} respectively at various complex numbers {1+it} with {|t| \leq T}. One can use {L^2} based estimates to control the Dirichlet series of {f_2,f_3}, while using the hypothesis (1) one can get {L^\infty} estimates on the Dirichlet series of {f_1}. (This is similar to the Fourier-analytic approach to ternary additive problems, such as Vinogradov’s theorem on representing large odd numbers as the sum of three primes.) This idea was inspired by a similar device used in the work of Granville, Harper, and Soundarajan. A variant of this argument also appears in unpublished work of Adam Harper.

I thank Andrew Granville for helpful comments which led to significant simplifications of the argument.

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The Chowla conjecture asserts, among other things, that one has the asymptotic

\displaystyle \frac{1}{X} \sum_{n \leq X} \lambda(n+h_1) \dots \lambda(n+h_k) = o(1)

as {X \rightarrow \infty} for any distinct integers {h_1,\dots,h_k}, where {\lambda} is the Liouville function. (The usual formulation of the conjecture also allows one to consider more general linear forms {a_i n + b_i} than the shifts {n+h_i}, but for sake of discussion let us focus on the shift case.) This conjecture remains open for {k \geq 2}, though there are now some partial results when one averages either in {x} or in the {h_1,\dots,h_k}, 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

\displaystyle \frac{1}{X} \sum_{n \leq X} g_1(n+h_1) \dots g_k(n+h_k) = o(1) \ \ \ \ \ (1)

whenever {g_1,\dots,g_k} were bounded completely multiplicative functions and {h_1,\dots,h_k} were distinct integers, and one of the {g_i} was “non-pretentious” in the sense that

\displaystyle \sum_p \frac{1 - \hbox{Re}( g_i(p) \overline{\chi(p)} p^{-it})}{p} = +\infty \ \ \ \ \ (2)

for all Dirichlet characters {\chi} and real numbers {t}. It is easy to see that some condition like (2) is necessary; for instance if {g(n) := \chi(n) n^{it}} and {\chi} has period {q} then {\frac{1}{X} \sum_{n \leq X} g(n+q) \overline{g(n)}} can be verified to be bounded away from zero as {X \rightarrow \infty}.

In a previous paper with Matomaki and Radziwill, we provided a counterexample to the original formulation of the Elliott conjecture, and proposed that (2) be replaced with the stronger condition

\displaystyle \inf_{|t| \leq X} \sum_{p \leq X} \frac{1 - \hbox{Re}( g_i(p) \overline{\chi(p)} p^{-it})}{p} \rightarrow +\infty \ \ \ \ \ (3)

as {X \rightarrow \infty} for any Dirichlet character {\chi}. To support this conjecture, we proved an averaged and non-asymptotic version of this conjecture which roughly speaking showed a bound of the form

\displaystyle \frac{1}{H^k} \sum_{h_1,\dots,h_k \leq H} |\frac{1}{X} \sum_{n \leq X} g_1(n+h_1) \dots g_k(n+h_k)| \leq \varepsilon

whenever {H} was an arbitrarily slowly growing function of {X}, {X} was sufficiently large (depending on {\varepsilon,k} and the rate at which {H} grows), and one of the {g_i} obeyed the condition

\displaystyle \inf_{|t| \leq AX} \sum_{p \leq X} \frac{1 - \hbox{Re}( g_i(p) \overline{\chi(p)} p^{-it})}{p} \geq A \ \ \ \ \ (4)

for some {A} that was sufficiently large depending on {k,\varepsilon}, and all Dirichlet characters {\chi} of period at most {A}. As further support of this conjecture, I recently established the bound

\displaystyle \frac{1}{\log \omega} |\sum_{X/\omega \leq n \leq X} \frac{g_1(n+h_1) g_2(n+h_2)}{n}| \leq \varepsilon

under the same hypotheses, where {\omega} is an arbitrarily slowly growing function of {X}.

In view of these results, it is tempting to conjecture that the condition (4) for one of the {g_i} should be sufficient to obtain the bound

\displaystyle |\frac{1}{X} \sum_{n \leq X} g_1(n+h_1) \dots g_k(n+h_k)| \leq \varepsilon

when {A} is large enough depending on {k,\varepsilon}. This may well be the case for {k=2}. However, the purpose of this blog post is to record a simple counterexample for {k>2}. Let’s take {k=3} for simplicity. Let {t_0} be a quantity much larger than {X} but much smaller than {X^2} (e.g. {t = X^{3/2}}), and set

\displaystyle g_1(n) := n^{it_0}; \quad g_2(n) := n^{-2it_0}; \quad g_3(n) := n^{it_0}.

For {X/2 \leq n \leq X}, Taylor expansion gives

\displaystyle (n+1)^{it} = n^{it_0} \exp( i t_0 / n ) + o(1)

and

\displaystyle (n+2)^{it} = n^{it_0} \exp( 2 i t_0 / n ) + o(1)

and hence

\displaystyle g_1(n) g_2(n+1) g_3(n+2) = 1 + o(1)

and hence

\displaystyle |\frac{1}{X} \sum_{X/2 \leq n \leq X} g_1(n) g_2(n+1) g_3(n+2)| \gg 1.

On the other hand one can easily verify that all of the {g_1,g_2,g_3} obey (4) (the restriction {|t| \leq AX} there prevents {t} from getting anywhere close to {t_0}). So it seems the correct non-asymptotic version of the Elliott conjecture is the following:

Conjecture 1 (Non-asymptotic Elliott conjecture) Let {k} be a natural number, and let {h_1,\dots,h_k} be integers. Let {\varepsilon > 0}, let {A} be sufficiently large depending on {k,\varepsilon,h_1,\dots,h_k}, and let {X} be sufficiently large depending on {k,\varepsilon,h_1,\dots,h_k,A}. Let {g_1,\dots,g_k} be bounded multiplicative functions such that for some {1 \leq i \leq k}, one has

\displaystyle \inf_{|t| \leq AX^{k-1}} \sum_{p \leq X} \frac{1 - \hbox{Re}( g_i(p) \overline{\chi(p)} p^{-it})}{p} \geq A

for all Dirichlet characters {\chi} of conductor at most {A}. Then

\displaystyle |\frac{1}{X} \sum_{n \leq X} g_1(n+h_1) \dots g_k(n+h_k)| \leq \varepsilon.

The {k=1} case of this conjecture follows from the work of Halasz; in my recent paper a logarithmically averaged version of the {k=2} case of this conjecture is established. The requirement to take {t} to be as large as {A X^{k-1}} 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).)

A similar subtlety arises when trying to control the maximal integral

\displaystyle \frac{1}{X} \int_X^{2X} \sup_\alpha \frac{1}{H} |\sum_{x \leq n \leq x+H} g(n) e(\alpha n)|\ dx. \ \ \ \ \ (5)

In my previous paper with Matomaki and Radziwill, we could show that easier expression

\displaystyle \frac{1}{X} \sup_\alpha \int_X^{2X} \frac{1}{H} |\sum_{x \leq n \leq x+H} g(n) e(\alpha n)|\ dx. \ \ \ \ \ (6)

was small (for {H} a slowly growing function of {X}) if {g} was bounded and completely multiplicative, and one had a condition of the form

\displaystyle \inf_{|t| \leq AX} \sum_{p \leq X} \frac{1 - \hbox{Re}( g(p) \overline{\chi(p)} p^{-it})}{p} \geq A \ \ \ \ \ (7)

for some large {A}. However, to obtain an analogous bound for (5) it now appears that one needs to strengthen the above condition to

\displaystyle \inf_{|t| \leq AX^2} \sum_{p \leq X} \frac{1 - \hbox{Re}( g(p) \overline{\chi(p)} p^{-it})}{p} \geq A

in order to address the counterexample in which {g(n) = n^{it_0}} for some {t_0} between {X} and {X^2}. This seems to suggest that proving (5) (which is closely related to the {k=3} 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.

I recently learned about a curious operation on square matrices known as sweeping, which is used in numerical linear algebra (particularly in applications to statistics), as a useful and more robust variant of the usual Gaussian elimination operations seen in undergraduate linear algebra courses. Given an {n \times n} matrix {A := (a_{ij})_{1 \leq i,j \leq n}} (with, say, complex entries) and an index {1 \leq k \leq n}, with the entry {a_{kk}} non-zero, the sweep {\hbox{Sweep}_k[A] = (\hat a_{ij})_{1 \leq i,j \leq n}} of {A} at {k} is the matrix given by the formulae

\displaystyle  \hat a_{ij} := a_{ij} - \frac{a_{ik} a_{kj}}{a_{kk}}

\displaystyle  \hat a_{ik} := \frac{a_{ik}}{a_{kk}}

\displaystyle  \hat a_{kj} := \frac{a_{kj}}{a_{kk}}

\displaystyle  \hat a_{kk} := \frac{-1}{a_{kk}}

for all {i,j \in \{1,\dots,n\} \backslash \{k\}}. Thus for instance if {k=1}, and {A} is written in block form as

\displaystyle  A = \begin{pmatrix} a_{11} & X \\ Y & B \end{pmatrix} \ \ \ \ \ (1)

for some {1 \times n-1} row vector {X}, {n-1 \times 1} column vector {Y}, and {n-1 \times n-1} minor {B}, one has

\displaystyle  \hbox{Sweep}_1[A] = \begin{pmatrix} -1/a_{11} & X / a_{11} \\ Y/a_{11} & B - a_{11}^{-1} YX \end{pmatrix}. \ \ \ \ \ (2)

The inverse sweep operation {\hbox{Sweep}_k^{-1}[A] = (\check a_{ij})_{1 \leq i,j \leq n}} is given by a nearly identical set of formulae:

\displaystyle  \check a_{ij} := a_{ij} - \frac{a_{ik} a_{kj}}{a_{kk}}

\displaystyle  \check a_{ik} := -\frac{a_{ik}}{a_{kk}}

\displaystyle  \check a_{kj} := -\frac{a_{kj}}{a_{kk}}

\displaystyle  \check a_{kk} := \frac{-1}{a_{kk}}

for all {i,j \in \{1,\dots,n\} \backslash \{k\}}. One can check that these operations invert each other. Actually, each sweep turns out to have order {4}, so that {\hbox{Sweep}_k^{-1} = \hbox{Sweep}_k^3}: an inverse sweep performs the same operation as three forward sweeps. Sweeps also preserve the space of symmetric matrices (allowing one to cut down computational run time in that case by a factor of two), and behave well with respect to principal minors; a sweep of a principal minor is a principal minor of a sweep, after adjusting indices appropriately.

Remarkably, the sweep operators all commute with each other: {\hbox{Sweep}_k \hbox{Sweep}_l = \hbox{Sweep}_l \hbox{Sweep}_k}. If {1 \leq k \leq n} and we perform the first {k} sweeps (in any order) to a matrix

\displaystyle  A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}

with {A_{11}} a {k \times k} minor, {A_{12}} a {k \times n-k} matrix, {A_{12}} a {n-k \times k} matrix, and {A_{22}} a {n-k \times n-k} matrix, one obtains the new matrix

\displaystyle  \hbox{Sweep}_1 \dots \hbox{Sweep}_k[A] = \begin{pmatrix} -A_{11}^{-1} & A_{11}^{-1} A_{12} \\ A_{21} A_{11}^{-1} & A_{22} - A_{21} A_{11}^{-1} A_{12} \end{pmatrix}.

Note the appearance of the Schur complement in the bottom right block. Thus, for instance, one can essentially invert a matrix {A} by performing all {n} sweeps:

\displaystyle  \hbox{Sweep}_1 \dots \hbox{Sweep}_n[A] = -A^{-1}.

If a matrix has the form

\displaystyle  A = \begin{pmatrix} B & X \\ Y & a \end{pmatrix}

for a {n-1 \times n-1} minor {B}, {n-1 \times 1} column vector {X}, {1 \times n-1} row vector {Y}, and scalar {a}, then performing the first {n-1} sweeps gives

\displaystyle  \hbox{Sweep}_1 \dots \hbox{Sweep}_{n-1}[A] = \begin{pmatrix} -B^{-1} & B^{-1} X \\ Y B^{-1} & a - Y B^{-1} X \end{pmatrix}

and all the components of this matrix are usable for various numerical linear algebra applications in statistics (e.g. in least squares regression). Given that sweeps behave well with inverses, it is perhaps not surprising that sweeps also behave well under determinants: the determinant of {A} can be factored as the product of the entry {a_{kk}} and the determinant of the {n-1 \times n-1} matrix formed from {\hbox{Sweep}_k[A]} by removing the {k^{th}} row and column. As a consequence, one can compute the determinant of {A} fairly efficiently (so long as the sweep operations don’t come close to dividing by zero) by sweeping the matrix for {k=1,\dots,n} in turn, and multiplying together the {kk^{th}} entry of the matrix just before the {k^{th}} sweep for {k=1,\dots,n} to obtain the determinant.

It turns out that there is a simple geometric explanation for these seemingly magical properties of the sweep operation. Any {n \times n} matrix {A} creates a graph {\hbox{Graph}[A] := \{ (X, AX): X \in {\bf R}^n \}} (where we think of {{\bf R}^n} as the space of column vectors). This graph is an {n}-dimensional subspace of {{\bf R}^n \times {\bf R}^n}. Conversely, most subspaces of {{\bf R}^n \times {\bf R}^n} arises as graphs; there are some that fail the vertical line test, but these are a positive codimension set of counterexamples.

We use {e_1,\dots,e_n,f_1,\dots,f_n} to denote the standard basis of {{\bf R}^n \times {\bf R}^n}, with {e_1,\dots,e_n} the standard basis for the first factor of {{\bf R}^n} and {f_1,\dots,f_n} the standard basis for the second factor. The operation of sweeping the {k^{th}} entry then corresponds to a ninety degree rotation {\hbox{Rot}_k: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}^n \times {\bf R}^n} in the {e_k,f_k} plane, that sends {f_k} to {e_k} (and {e_k} to {-f_k}), keeping all other basis vectors fixed: thus we have

\displaystyle  \hbox{Graph}[ \hbox{Sweep}_k[A] ] = \hbox{Rot}_k \hbox{Graph}[A]

for generic {n \times n} {A} (more precisely, those {A} with non-vanishing entry {a_{kk}}). For instance, if {k=1} and {A} is of the form (1), then {\hbox{Graph}[A]} is the set of tuples {(r,R,s,S) \in {\bf R} \times {\bf R}^{n-1} \times {\bf R} \times {\bf R}^{n-1}} obeying the equations

\displaystyle  a_{11} r + X R = s

\displaystyle  Y r + B R = S.

The image of {(r,R,s,S)} under {\hbox{Rot}_1} is {(s, R, -r, S)}. Since we can write the above system of equations (for {a_{11} \neq 0}) as

\displaystyle  \frac{-1}{a_{11}} s + \frac{X}{a_{11}} R = -r

\displaystyle  \frac{Y}{a_{11}} s + (B - a_{11}^{-1} YX) R = S

we see from (2) that {\hbox{Rot}_1 \hbox{Graph}[A]} is the graph of {\hbox{Sweep}_1[A]}. Thus the sweep operation is a multidimensional generalisation of the high school geometry fact that the line {y = mx} in the plane becomes {y = \frac{-1}{m} x} after applying a ninety degree rotation.

It is then an instructive exercise to use this geometric interpretation of the sweep operator to recover all the remarkable properties about these operations listed above. It is also useful to compare the geometric interpretation of sweeping as rotation of the graph to that of Gaussian elimination, which instead shears and reflects the graph by various elementary transformations (this is what is going on geometrically when one performs Gaussian elimination on an augmented matrix). Rotations are less distorting than shears, so one can see geometrically why sweeping can produce fewer numerical artefacts than Gaussian elimination.

Let {X} and {Y} be two random variables taking values in the same (discrete) range {R}, and let {E} be some subset of {R}, which we think of as the set of “bad” outcomes for either {X} or {Y}. If {X} and {Y} have the same probability distribution, then clearly

\displaystyle  {\bf P}( X \in E ) = {\bf P}( Y \in E ).

In particular, if it is rare for {Y} to lie in {E}, then it is also rare for {X} to lie in {E}.

If {X} and {Y} do not have exactly the same probability distribution, but their probability distributions are close to each other in some sense, then we can expect to have an approximate version of the above statement. For instance, from the definition of the total variation distance {\delta(X,Y)} between two random variables (or more precisely, the total variation distance between the probability distributions of two random variables), we see that

\displaystyle  {\bf P}(Y \in E) - \delta(X,Y) \leq {\bf P}(X \in E) \leq {\bf P}(Y \in E) + \delta(X,Y) \ \ \ \ \ (1)

for any {E \subset R}. In particular, if it is rare for {Y} to lie in {E}, and {X,Y} are close in total variation, then it is also rare for {X} to lie in {E}.

A basic inequality in information theory is Pinsker’s inequality

\displaystyle  \delta(X,Y) \leq \sqrt{\frac{1}{2} D_{KL}(X||Y)}

where the Kullback-Leibler divergence {D_{KL}(X||Y)} is defined by the formula

\displaystyle  D_{KL}(X||Y) = \sum_{x \in R} {\bf P}( X=x ) \log \frac{{\bf P}(X=x)}{{\bf P}(Y=x)}.

(See this previous blog post for a proof of this inequality.) A standard application of Jensen’s inequality reveals that {D_{KL}(X||Y)} is non-negative (Gibbs’ inequality), and vanishes if and only if {X}, {Y} have the same distribution; thus one can think of {D_{KL}(X||Y)} as a measure of how close the distributions of {X} and {Y} are to each other, although one should caution that this is not a symmetric notion of distance, as {D_{KL}(X||Y) \neq D_{KL}(Y||X)} in general. Inserting Pinsker’s inequality into (1), we see for instance that

\displaystyle  {\bf P}(X \in E) \leq {\bf P}(Y \in E) + \sqrt{\frac{1}{2} D_{KL}(X||Y)}.

Thus, if {X} is close to {Y} in the Kullback-Leibler sense, and it is rare for {Y} to lie in {E}, then it is rare for {X} to lie in {E} as well.

We can specialise this inequality to the case when {Y} a uniform random variable {U} on a finite range {R} of some cardinality {N}, in which case the Kullback-Leibler divergence {D_{KL}(X||U)} simplifies to

\displaystyle  D_{KL}(X||U) = \log N - {\bf H}(X)

where

\displaystyle  {\bf H}(X) := \sum_{x \in R} {\bf P}(X=x) \log \frac{1}{{\bf P}(X=x)}

is the Shannon entropy of {X}. Again, a routine application of Jensen’s inequality shows that {{\bf H}(X) \leq \log N}, with equality if and only if {X} is uniformly distributed on {R}. The above inequality then becomes

\displaystyle  {\bf P}(X \in E) \leq {\bf P}(U \in E) + \sqrt{\frac{1}{2}(\log N - {\bf H}(X))}. \ \ \ \ \ (2)

Thus, if {E} is a small fraction of {R} (so that it is rare for {U} to lie in {E}), and the entropy of {X} is very close to the maximum possible value of {\log N}, then it is rare for {X} to lie in {E} also.

The inequality (2) is only useful when the entropy {{\bf H}(X)} is close to {\log N} in the sense that {{\bf H}(X) = \log N - O(1)}, otherwise the bound is worse than the trivial bound of {{\bf P}(X \in E) \leq 1}. In my recent paper on the Chowla and Elliott conjectures, I ended up using a variant of (2) which was still non-trivial when the entropy {{\bf H}(X)} was allowed to be smaller than {\log N - O(1)}. More precisely, I used the following simple inequality, which is implicit in the arguments of that paper but which I would like to make more explicit in this post:

Lemma 1 (Pinsker-type inequality) Let {X} be a random variable taking values in a finite range {R} of cardinality {N}, let {U} be a uniformly distributed random variable in {R}, and let {E} be a subset of {R}. Then

\displaystyle  {\bf P}(X \in E) \leq \frac{(\log N - {\bf H}(X)) + \log 2}{\log 1/{\bf P}(U \in E)}.

Proof: Consider the conditional entropy {{\bf H}(X | 1_{X \in E} )}. On the one hand, we have

\displaystyle  {\bf H}(X | 1_{X \in E} ) = {\bf H}(X, 1_{X \in E}) - {\bf H}(1_{X \in E} )

\displaystyle  = {\bf H}(X) - {\bf H}(1_{X \in E})

\displaystyle  \geq {\bf H}(X) - \log 2

by Jensen’s inequality. On the other hand, one has

\displaystyle  {\bf H}(X | 1_{X \in E} ) = {\bf P}(X \in E) {\bf H}(X | X \in E )

\displaystyle  + (1-{\bf P}(X \in E)) {\bf H}(X | X \not \in E)

\displaystyle  \leq {\bf P}(X \in E) \log |E| + (1-{\bf P}(X \in E)) \log N

\displaystyle  = \log N - {\bf P}(X \in E) \log \frac{N}{|E|}

\displaystyle  = \log N - {\bf P}(X \in E) \log \frac{1}{{\bf P}(U \in E)},

where we have again used Jensen’s inequality. Putting the two inequalities together, we obtain the claim. \Box

Remark 2 As noted in comments, this inequality can be viewed as a special case of the more general inequality

\displaystyle  {\bf P}(X \in E) \leq \frac{D(X||Y) + \log 2}{\log 1/{\bf P}(Y \in E)}

for arbitrary random variables {X,Y} taking values in the same discrete range {R}, which follows from the data processing inequality

\displaystyle  D( f(X)||f(Y)) \leq D(X|| Y)

for arbitrary functions {f}, applied to the indicator function {f = 1_E}. Indeed one has

\displaystyle  D( 1_E(X) || 1_E(Y) ) = {\bf P}(X \in E) \log \frac{{\bf P}(X \in E)}{{\bf P}(Y \in E)}

\displaystyle + {\bf P}(X \not \in E) \log \frac{{\bf P}(X \not \in E)}{{\bf P}(Y \not \in E)}

\displaystyle  \geq {\bf P}(X \in E) \log \frac{1}{{\bf P}(Y \in E)} - h( {\bf P}(X \in E) )

\displaystyle  \geq {\bf P}(X \in E) \log \frac{1}{{\bf P}(Y \in E)} - \log 2

where {h(u) := u \log \frac{1}{u} + (1-u) \log \frac{1}{1-u}} is the entropy function.

Thus, for instance, if one has

\displaystyle  {\bf H}(X) \geq \log N - o(K)

and

\displaystyle  {\bf P}(U \in E) \leq \exp( - K )

for some {K} much larger than {1} (so that {1/K = o(1)}), then

\displaystyle  {\bf P}(X \in E) = o(1).

More informally: if the entropy of {X} is somewhat close to the maximum possible value of {\log N}, and it is exponentially rare for a uniform variable to lie in {E}, then it is still somewhat rare for {X} to lie in {E}. The estimate given is close to sharp in this regime, as can be seen by calculating the entropy of a random variable {X} which is uniformly distributed inside a small set {E} with some probability {p} and uniformly distributed outside of {E} with probability {1-p}, for some parameter {0 \leq p \leq 1}.

It turns out that the above lemma combines well with concentration of measure estimates; in my paper, I used one of the simplest such estimates, namely Hoeffding’s inequality, but there are of course many other estimates of this type (see e.g. this previous blog post for some others). Roughly speaking, concentration of measure inequalities allow one to make approximations such as

\displaystyle  F(U) \approx {\bf E} F(U)

with exponentially high probability, where {U} is a uniform distribution and {F} is some reasonable function of {U}. Combining this with the above lemma, we can then obtain approximations of the form

\displaystyle  F(X) \approx {\bf E} F(U) \ \ \ \ \ (3)

with somewhat high probability, if the entropy of {X} is somewhat close to maximum. This observation, combined with an “entropy decrement argument” that allowed one to arrive at a situation in which the relevant random variable {X} did have a near-maximum entropy, is the key new idea in my recent paper; for instance, one can use the approximation (3) to obtain an approximation of the form

\displaystyle  \sum_{j=1}^H \sum_{p \in {\mathcal P}} \lambda(n+j) \lambda(n+j+p) 1_{p|n+j}

\displaystyle  \approx \sum_{j=1}^H \sum_{p \in {\mathcal P}} \frac{\lambda(n+j) \lambda(n+j+p)}{p}

for “most” choices of {n} and a suitable choice of {H} (with the latter being provided by the entropy decrement argument). The left-hand side is tied to Chowla-type sums such as {\sum_{n \leq x} \frac{\lambda(n)\lambda(n+1)}{n}} through the multiplicativity of {\lambda}, while the right-hand side, being a linear correlation involving two parameters {j,p} rather than just one, has “finite complexity” and can be treated by existing techniques such as the Hardy-Littlewood circle method. One could hope that one could similarly use approximations such as (3) in other problems in analytic number theory or combinatorics.

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