You are currently browsing the category archive for the ‘Mathematics’ category.

Brad Rodgers and I have uploaded to the arXiv our paper “The De Bruijn-Newman constant is non-negative“. This paper affirms a conjecture of Newman regarding to the extent to which the Riemann hypothesis, if true, is only “barely so”. To describe the conjecture, let us begin with the Riemann xi function

where is the Gamma function and is the Riemann zeta function. Initially, this function is only defined for , but, as was already known to Riemann, we can manipulate it into a form that extends to the entire complex plane as follows. Firstly, in view of the standard identity , we can write

and hence

By a rescaling, one may write

and similarly

and thus (after applying Fubini’s theorem)

We’ll make the change of variables to obtain

If we introduce the mild renormalisation

of , we then conclude (at least for ) that

which one can verify to be rapidly decreasing both as and as , with the decrease as faster than any exponential. In particular extends holomorphically to the upper half plane.

If we normalize the Fourier transform of a (Schwartz) function as , it is well known that the Gaussian is its own Fourier transform. The creation operator interacts with the Fourier transform by the identity

Since , this implies that the function

is its own Fourier transform. (One can view the polynomial as a renormalised version of the fourth Hermite polynomial.) Taking a suitable linear combination of this with , we conclude that

is also its own Fourier transform. Rescaling by and then multiplying by , we conclude that the Fourier transform of

is

and hence by the Poisson summation formula (using symmetry and vanishing at to unfold the summation in (2) to the integers rather than the natural numbers) we obtain the functional equation

which implies that and are even functions (in particular, now extends to an entire function). From this symmetry we can also rewrite (1) as

which now gives a convergent expression for the entire function for all complex . As is even and real-valued on , is even and also obeys the functional equation , which is equivalent to the usual functional equation for the Riemann zeta function. The Riemann hypothesis is equivalent to the claim that all the zeroes of are real.

De Bruijn introduced the family of deformations of , defined for all and by the formula

From a PDE perspective, one can view as the evolution of under the backwards heat equation . As with , the are all even entire functions that obey the functional equation , and one can ask an analogue of the Riemann hypothesis for each such , namely whether all the zeroes of are real. De Bruijn showed that these hypotheses were monotone in : if had all real zeroes for some , then would also have all zeroes real for any . Newman later sharpened this claim by showing the existence of a finite number , now known as the *de Bruijn-Newman constant*, with the property that had all zeroes real if and only if . Thus, the Riemann hypothesis is equivalent to the inequality . Newman then conjectured the complementary bound ; in his words, this conjecture asserted that if the Riemann hypothesis is true, then it is only “barely so”, in that the reality of all the zeroes is destroyed by applying heat flow for even an arbitrarily small amount of time. Over time, a significant amount of evidence was established in favour of this conjecture; most recently, in 2011, Saouter, Gourdon, and Demichel showed that .

In this paper we finish off the proof of Newman’s conjecture, that is we show that . The proof is by contradiction, assuming that (which among other things, implies the truth of the Riemann hypothesis), and using the properties of backwards heat evolution to reach a contradiction.

Very roughly, the argument proceeds as follows. As observed by Csordas, Smith, and Varga (and also discussed in this previous blog post, the backwards heat evolution of the introduces a nice ODE dynamics on the zeroes of , namely that they solve the ODE

for all (one has to interpret the sum in a principal value sense as it is not absolutely convergent, but let us ignore this technicality for the current discussion). Intuitively, this ODE is asserting that the zeroes repel each other, somewhat like positively charged particles (but note that the dynamics is first-order, as opposed to the second-order laws of Newtonian mechanics). Formally, a steady state (or equilibrium) of this dynamics is reached when the are arranged in an arithmetic progression. (Note for instance that for any positive , the functions obey the same backwards heat equation as , and their zeroes are on a fixed arithmetic progression .) The strategy is to then show that the dynamics from time to time creates a *convergence to local equilibrium*, in which the zeroes locally resemble an arithmetic progression at time . This will be in contradiction with known results on pair correlation of zeroes (or on related statistics, such as the fluctuations on gaps between zeroes), such as the results of Montgomery (actually for technical reasons it is slightly more convenient for us to use related results of Conrey, Ghosh, Goldston, Gonek, and Heath-Brown). Another way of thinking about this is that even very slight deviations from local equilibrium (such as a small number of gaps that are slightly smaller than the average spacing) will almost immediately lead to zeroes colliding with each other and leaving the real line as one evolves backwards in time (i.e., under the *forward* heat flow). This is a refinement of the strategy used in previous lower bounds on , in which “Lehmer pairs” (pairs of zeroes of the zeta function that were unusually close to each other) were used to limit the extent to which the evolution continued backwards in time while keeping all zeroes real.

How does one obtain this convergence to local equilibrium? We proceed by broad analogy with the “local relaxation flow” method of Erdos, Schlein, and Yau in random matrix theory, in which one combines some initial control on zeroes (which, in the case of the Erdos-Schlein-Yau method, is referred to with terms such as “local semicircular law”) with convexity properties of a relevant Hamiltonian that can be used to force the zeroes towards equilibrium.

We first discuss the initial control on zeroes. For , we have the classical Riemann-von Mangoldt formula, which asserts that the number of zeroes in the interval is as . (We have a factor of here instead of the more familiar due to the way is normalised.) This implies for instance that for a fixed , the number of zeroes in the interval is . Actually, because we get to assume the Riemann hypothesis, we can sharpen this to , a result of Littlewood (see this previous blog post for a proof). Ideally, we would like to obtain similar control for the other , , as well. Unfortunately we were only able to obtain the weaker claims that the number of zeroes of in is , and that the number of zeroes in is , that is to say we only get good control on the distribution of zeroes at scales rather than at scales . Ultimately this is because we were only able to get control (and in particular, lower bounds) on with high precision when (whereas has good estimates as soon as is larger than (say) ). This control is obtained by the expressing in terms of some contour integrals and using the method of steepest descent (actually it is slightly simpler to rely instead on the Stirling approximation for the Gamma function, which can be proven in turn by steepest descent methods). Fortunately, it turns out that this weaker control is still (barely) enough for the rest of our argument to go through.

Once one has the initial control on zeroes, we now need to force convergence to local equilibrium by exploiting convexity of a Hamiltonian. Here, the relevant Hamiltonian is

ignoring for now the rather important technical issue that this sum is not actually absolutely convergent. (Because of this, we will need to truncate and renormalise the Hamiltonian in a number of ways which we will not detail here.) The ODE (3) is formally the gradient flow for this Hamiltonian. Furthermore, this Hamiltonian is a convex function of the (because is a convex function on ). We therefore expect the Hamiltonian to be a decreasing function of time, and that the derivative should be an increasing function of time. As time passes, the derivative of the Hamiltonian would then be expected to converge to zero, which should imply convergence to local equilibrium.

Formally, the derivative of the above Hamiltonian is

Again, there is the important technical issue that this quantity is infinite; but it turns out that if we renormalise the Hamiltonian appropriately, then the energy will also become suitably renormalised, and in particular will vanish when the are arranged in an arithmetic progression, and be positive otherwise. One can also formally calculate the derivative of to be a somewhat complicated but manifestly non-negative quantity (a sum of squares); see this previous blog post for analogous computations in the case of heat flow on polynomials. After flowing from time to time , and using some crude initial bounds on and in this region (coming from the Riemann-von Mangoldt type formulae mentioned above and some further manipulations), we can eventually show that the (renormalisation of the) energy at time zero is small, which forces the to locally resemble an arithmetic progression, which gives the required convergence to local equilibrium.

There are a number of technicalities involved in making the above sketch of argument rigorous (for instance, justifying interchanges of derivatives and infinite sums turns out to be a little bit delicate). I will highlight here one particular technical point. One of the ways in which we make expressions such as the energy finite is to truncate the indices to an interval to create a truncated energy . In typical situations, we would then expect to be decreasing, which will greatly help in bounding (in particular it would allow one to control by time-averaged quantities such as , which can in turn be controlled using variants of (4)). However, there are boundary effects at both ends of that could in principle add a large amount of energy into , which is bad news as it could conceivably make undesirably large even if integrated energies such as remain adequately controlled. As it turns out, such boundary effects are negligible as long as there is a large gap between adjacent zeroes at boundary of – it is only narrow gaps that can rapidly transmit energy across the boundary of . Now, narrow gaps can certainly exist (indeed, the GUE hypothesis predicts these happen a positive fraction of the time); but the pigeonhole principle (together with the Riemann-von Mangoldt formula) can allow us to pick the endpoints of the interval so that no narrow gaps appear at the boundary of for any given time . However, there was a technical problem: this argument did not allow one to find a single interval that avoided gaps for *all* times simultaneously – the pigeonhole principle could produce a different interval for each time ! Since the number of times was uncountable, this was a serious issue. (In physical terms, the problem was that there might be very fast “longitudinal waves” in the dynamics that, at each time, cause some gaps between zeroes to be highly compressed, but the specific gap that was narrow changed very rapidly with time. Such waves could, in principle, import a huge amount of energy into by time .) To resolve this, we borrowed a PDE trick of Bourgain’s, in which the pigeonhole principle was coupled with local conservation laws. More specifically, we use the phenomenon that very narrow gaps take a nontrivial amount of time to expand back to a reasonable size (this can be seen by comparing the evolution of this gap with solutions of the scalar ODE , which represents the fastest at which a gap such as can expand). Thus, if a gap is reasonably large at some time , it will also stay reasonably large at slightly earlier times for some moderately small . This lets one locate an interval that has manageable boundary effects during the times in , so in particular is basically non-increasing in this time interval. Unfortunately, this interval is a little bit too short to cover all of ; however it turns out that one can iterate the above construction and find a nested sequence of intervals , with each non-increasing in a different time interval , and with all of the time intervals covering . This turns out to be enough (together with the obvious fact that is monotone in ) to still control for some reasonably sized interval , as required for the rest of the arguments.

ADDED LATER: the following analogy (involving functions with just two zeroes, rather than an infinite number of zeroes) may help clarify the relation between this result and the Riemann hypothesis (and in particular why this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis). Suppose one had a quadratic polynomial of the form , where was an unknown real constant. Suppose that one was for some reason interested in the analogue of the “Riemann hypothesis” for , namely that all the zeroes of are real. A priori, there are three scenarios:

- (Riemann hypothesis false) , and has zeroes off the real axis.
- (Riemann hypothesis true, but barely so) , and both zeroes of are on the real axis; however, any slight perturbation of in the positive direction would move zeroes off the real axis.
- (Riemann hypothesis true, with room to spare) , and both zeroes of are on the real axis. Furthermore, any slight perturbation of will also have both zeroes on the real axis.

The analogue of our result in this case is that , thus ruling out the third of the three scenarios here. In this simple example in which only two zeroes are involved, one can think of the inequality as asserting that if the zeroes of are real, then they must be repeated. In our result (in which there are an infinity of zeroes, that become increasingly dense near infinity), and in view of the convergence to local equilibrium properties of (3), the analogous assertion is that if the zeroes of are real, then they do not behave locally as if they were in arithmetic progression.

The Polymath14 online collaboration has uploaded to the arXiv its paper “Homogeneous length functions on groups“, submitted to Algebra & Number Theory. The paper completely classifies *homogeneous length functions* on an arbitrary group , that is to say non-negative functions that obey the symmetry condition , the non-degeneracy condition , the triangle inequality , and the homogeneity condition . It turns out that these norms can only arise from pulling back the norm of a Banach space by an isometric embedding of the group. Among other things, this shows that can only support a homogeneous length function if and only if it is abelian and torsion free, thus giving a metric description of this property.

The proof is based on repeated use of the homogeneous length function axioms, combined with elementary identities of commutators, to obtain increasingly good bounds on quantities such as , until one can show that such norms have to vanish. See the previous post for a full proof. The result is robust in that it allows for some loss in the triangle inequality and homogeneity condition, allowing for some new results on “quasinorms” on groups that relate to quasihomomorphisms.

As there are now a large number of comments on the previous post on this project, this post will also serve as the new thread for any final discussion of this project as it winds down.

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges“. This is a sequel of sorts to our previous paper on divisor correlations, though the proof techniques in this paper are rather different. As with the previous paper, our interest is in correlations such as

for medium-sized and large , where are natural numbers and is the divisor function (actually our methods can also treat a generalisation in which is non-integer, but for simplicity let us stick with the integer case for this discussion). Our methods also allow for one of the divisor function factors to be replaced with a von Mangoldt function, but (in contrast to the previous paper) we cannot treat the case when both factors are von Mangoldt.

As discussed in this previous post, one heuristically expects an asymptotic of the form

for any fixed , where is a certain explicit (but rather complicated) polynomial of degree . Such asymptotics are known when , but remain open for . In the previous paper, we were able to obtain a weaker bound of the form

for of the shifts , whenever the shift range lies between and . But the methods become increasingly hard to use as gets smaller. In this paper, we use a rather different method to obtain the even weaker bound

for of the shifts , where can now be as short as . The constant can be improved, but there are serious obstacles to using our method to go below (as the exceptionally large values of then begin to dominate). This can be viewed as an analogue to our previous paper on correlations of bounded multiplicative functions on average, in which the functions are now unbounded, and indeed our proof strategy is based in large part on that paper (but with many significant new technical complications).

We now discuss some of the ingredients of the proof. Unsurprisingly, the first step is the circle method, expressing (1) in terms of exponential sums such as

Actually, it is convenient to first prune slightly by zeroing out this function on “atypical” numbers that have an unusually small or large number of factors in a certain sense, but let us ignore this technicality for this discussion. The contribution of for “major arc” can be treated by standard techniques (and is the source of the main term ; the main difficulty comes from treating the contribution of “minor arc” .

In our previous paper on bounded multiplicative functions, we used Plancherel’s theorem to estimate the global norm , and then also used the Katai-Bourgain-Sarnak-Ziegler orthogonality criterion to control local norms , where was a minor arc interval of length about , and these two estimates together were sufficient to get a good bound on correlations by an application of Hölder’s inequality. For , it is more convenient to use Dirichlet series methods (and Ramaré-type factorisations of such Dirichlet series) to control local norms on minor arcs, in the spirit of the proof of the Matomaki-Radziwill theorem; a key point is to develop “log-free” mean value theorems for Dirichlet series associated to functions such as , so as not to wipe out the (rather small) savings one will get over the trivial bound from this method. On the other hand, the global bound will definitely be unusable, because the sum has too many unwanted factors of . Fortunately, we can substitute this global bound with a “large values” bound that controls expressions such as

for a moderate number of disjoint intervals , with a bound that is slightly better (for a medium-sized power of ) than what one would have obtained by bounding each integral separately. (One needs to save more than for the argument to work; we end up saving a factor of about .) This large values estimate is probably the most novel contribution of the paper. After taking the Fourier transform, matters basically reduce to getting a good estimate for

where is the midpoint of ; thus we need some upper bound on the large local Fourier coefficients of . These coefficients are difficult to calculate directly, but, in the spirit of a paper of Ben Green and myself, we can try to replace by a more tractable and “pseudorandom” majorant for which the local Fourier coefficients are computable (on average). After a standard duality argument, one ends up having to control expressions such as

after various averaging in the parameters. These local Fourier coefficients of turn out to be small on average unless is “major arc”. One then is left with a mostly combinatorial problem of trying to bound how often this major arc scenario occurs. This is very close to a computation in the previously mentioned paper of Ben and myself; there is a technical wrinkle in that the are not as well separated as they were in my paper with Ben, but it turns out that one can modify the arguments in that paper to still obtain a satisfactory estimate in this case (after first grouping nearby frequencies together, and modifying the duality argument accordingly).

In the tradition of “Polymath projects“, the problem posed in the previous two blog posts has now been solved, thanks to the cumulative effect of many small contributions by many participants (including, but not limited to, Sean Eberhard, Tobias Fritz, Siddharta Gadgil, Tobias Hartnick, Chris Jerdonek, Apoorva Khare, Antonio Machiavelo, Pace Nielsen, Andy Putman, Will Sawin, Alexander Shamov, Lior Silberman, and David Speyer). In this post I’ll write down a streamlined resolution, eliding a number of important but ultimately removable partial steps and insights made by the above contributors en route to the solution.

Theorem 1Let be a group. Suppose one has a “seminorm” function which obeys the triangle inequalityfor all , with equality whenever . Then the seminorm factors through the abelianisation map .

*Proof:* By the triangle inequality, it suffices to show that for all , where is the commutator.

We first establish some basic facts. Firstly, by hypothesis we have , and hence whenever is a power of two. On the other hand, by the triangle inequality we have for all positive , and hence by the triangle inequality again we also have the matching lower bound, thus

for all . The claim is also true for (apply the preceding bound with and ). By replacing with if necessary we may now also assume without loss of generality that , thus

Next, for any , and any natural number , we have

so on taking limits as we have . Replacing by gives the matching lower bound, thus we have the conjugation invariance

Next, we observe that if are such that is conjugate to both and , then one has the inequality

Indeed, if we write for some , then for any natural number one has

where the and terms each appear times. From (2) we see that conjugation by does not affect the norm. Using this and the triangle inequality several times, we conclude that

and the claim (3) follows by sending .

The following special case of (3) will be of particular interest. Let , and for any integers , define the quantity

Observe that is conjugate to both and to , hence by (3) one has

which by (2) leads to the recursive inequality

We can write this in probabilistic notation as

where is a random vector that takes the values and with probability each. Iterating this, we conclude in particular that for any large natural number , one has

where and are iid copies of . We can write where are iid signs. By the triangle inequality, we thus have

noting that is an even integer. On the other hand, has mean zero and variance , hence by Cauchy-Schwarz

But by (1), the left-hand side is equal to . Dividing by and then sending , we obtain the claim.

The above theorem reduces such seminorms to abelian groups. It is easy to see from (1) that any torsion element of such groups has zero seminorm, so we can in fact restrict to torsion-free groups, which we now write using additive notation , thus for instance for . We think of as a -module. One can then extend the seminorm to the associated -vector space by the formula , and then to the associated -vector space by continuity, at which point it becomes a genuine seminorm (provided we have ensured the symmetry condition ). Conversely, any seminorm on induces a seminorm on . (These arguments also appear in this paper of Khare and Rajaratnam.)

This post is a continuation of the previous post, which has attracted a large number of comments. I’m recording here some calculations that arose from those comments (particularly those of Pace Nielsen, Lior Silberman, Tobias Fritz, and Apoorva Khare). Please feel free to either continue these calculations or to discuss other approaches to the problem, such as those mentioned in the remaining comments to the previous post.

Let be the free group on two generators , and let be a quantity obeying the triangle inequality

and the linear growth property

for all and integers ; this implies the conjugation invariance

or equivalently

We consider inequalities of the form

for various real numbers . For instance, since

we have (1) for . We also have the following further relations:

Proposition 1

*Proof:* For (i) we simply observe that

For (ii), we calculate

giving the claim.

For (iii), we calculate

giving the claim.

For (iv), we calculate

giving the claim.

Here is a typical application of the above estimates. If (1) holds for , then by part (i) it holds for , then by (ii) (2) holds for , then by (iv) (1) holds for . The map has fixed point , thus

For instance, if , then .

Here is a curious question posed to me by Apoorva Khare that I do not know the answer to. Let be the free group on two generators . Does there exist a metric on this group which is

- bi-invariant, thus for all ; and
- linear growth in the sense that for all and all natural numbers ?

By defining the “norm” of an element to be , an equivalent formulation of the problem asks if there exists a non-negative norm function that obeys the conjugation invariance

for all , the triangle inequality

for all , and the linear growth

for all and , and such that for all non-identity . Indeed, if such a norm exists then one can just take to give the desired metric.

One can normalise the norm of the generators to be at most , thus

This can then be used to upper bound the norm of other words in . For instance, from (1), (3) one has

A bit less trivially, from (3), (2), (1) one can bound commutators as

In a similar spirit one has

What is not clear to me is if one can keep arguing like this to continually improve the upper bounds on the norm of a given non-trivial group element to the point where this norm must in fact vanish, which would demonstrate that no metric with the above properties on would exist (and in fact would impose strong constraints on similar metrics existing on other groups as well). It is also tempting to use some ideas from geometric group theory (e.g. asymptotic cones) to try to understand these metrics further, though I wasn’t able to get very far with this approach. Anyway, this feels like a problem that might be somewhat receptive to a more crowdsourced attack, so I am posing it here in case any readers wish to try to make progress on it.

The Boussinesq equations for inviscid, incompressible two-dimensional fluid flow in the presence of gravity are given by

where is the velocity field, is the pressure field, and is the density field (or, in some physical interpretations, the temperature field). In this post we shall restrict ourselves to formal manipulations, assuming implicitly that all fields are regular enough (or sufficiently decaying at spatial infinity) that the manipulations are justified. Using the material derivative , one can abbreviate these equations as

One can eliminate the role of the pressure by working with the *vorticity* . A standard calculation then leads us to the equivalent “vorticity-stream” formulation

of the Boussinesq equations. The latter two equations can be used to recover the velocity field from the vorticity by the Biot-Savart law

It has long been observed (see e.g. Section 5.4.1 of Bertozzi-Majda) that the Boussinesq equations are very similar, though not quite identical, to the three-dimensional inviscid incompressible Euler equations under the hypothesis of axial symmetry (with swirl). The Euler equations are

where now the velocity field and pressure field are over the three-dimensional domain . If one expresses in polar coordinates then one can write the velocity vector field in these coordinates as

If we make the axial symmetry assumption that these components, as well as , do not depend on the variable, thus

then after some calculation (which we give below the fold) one can eventually reduce the Euler equations to the system

where is the modified material derivative, and is the field . This is almost identical with the Boussinesq equations except for some additional powers of ; thus, the intuition is that the Boussinesq equations are a simplified model for axially symmetric Euler flows when one stays away from the axis and also does not wander off to .

However, this heuristic is not rigorous; the above calculations do not actually give an embedding of the Boussinesq equations into Euler. (The equations do match on the cylinder , but this is a measure zero subset of the domain, and so is not enough to give an embedding on any non-trivial region of space.) Recently, while playing around with trying to embed other equations into the Euler equations, I discovered that it is possible to make such an embedding into a *four*-dimensional Euler equation, albeit on a slightly curved manifold rather than in Euclidean space. More precisely, we use the Ebin-Marsden generalisation

of the Euler equations to an arbitrary Riemannian manifold (ignoring any issues of boundary conditions for this discussion), where is a time-dependent vector field, is a time-dependent scalar field, and is the covariant derivative along using the Levi-Civita connection . In Penrose abstract index notation (using the Levi-Civita connection , and raising and lowering indices using the metric ), the equations of motion become

in coordinates, this becomes

where the Christoffel symbols are given by the formula

where is the inverse to the metric tensor . If the coordinates are chosen so that the volume form is the Euclidean volume form , thus , then on differentiating we have , and hence , and so the divergence-free equation (10) simplifies in this case to . The Ebin-Marsden Euler equations are the natural generalisation of the Euler equations to arbitrary manifolds; for instance, they (formally) conserve the kinetic energy

and can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on (see this previous post for a discussion of this in the flat space case).

The specific four-dimensional manifold in question is the space with metric

and solutions to the Boussinesq equation on can be transformed into solutions to the Euler equations on this manifold. This is part of a more general family of embeddings into the Euler equations in which passive scalar fields (such as the field appearing in the Boussinesq equations) can be incorporated into the dynamics via fluctuations in the Riemannian metric ). I am writing the details below the fold (partly for my own benefit).

I have just uploaded to the arXiv the paper “An inverse theorem for an inequality of Kneser“, submitted to a special issue of the Proceedings of the Steklov Institute of Mathematics in honour of Sergei Konyagin. It concerns an inequality of Kneser discussed previously in this blog, namely that

whenever are compact non-empty subsets of a compact connected additive group with probability Haar measure . (A later result of Kemperman extended this inequality to the nonabelian case.) This inequality is non-trivial in the regime

The connectedness of is essential, otherwise one could form counterexamples involving proper subgroups of of positive measure. In the blog post, I indicated how this inequality (together with a more “robust” strengthening of it) could be deduced from submodularity inequalities such as

which in turn easily follows from the identity and the inclusion , combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking to be the unit circle and to be arcs in that circle (obeying (2)). A bit more generally, if is an arbitrary connected compact abelian group and is a non-trivial character (i.e., a continuous homomorphism), then must be surjective (as has no non-trivial connected subgroups), and one can take and for some arcs in that circle (again choosing the measures of these arcs to obey (2)). The main result of this paper is an inverse theorem that asserts that this is the only way in which equality can occur in (1) (assuming (2)); furthermore, if (1) is close to being satisfied with equality and (2) holds, then must be close (in measure) to an example of the above form . Actually, for technical reasons (and for the applications we have in mind), it is important to establish an inverse theorem not just for (1), but for the more robust version mentioned earlier (in which the sumset is replaced by the partial sumset consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call a *critical pair* if (2) holds and the inequality (1) (or more precisely, a robust version of this inequality) is almost obeyed with equality. The notion of a critical pair obeys some useful closure properties. Firstly, it is symmetric in , and invariant with respect to translation of either or . Furthermore, from the submodularity inequality (3), one can show that if and are critical pairs (with and positive), then and are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when come from arcs of a circle.) Similarly, from associativity , one can show that if and are critical pairs, then so are and .

One can combine these closure properties to obtain further ones. For instance, suppose is such that . Then (cheating a little bit), one can show that is also a critical pair, basically because is the union of the , , the are all critical pairs, and the all intersect each other. This argument doesn’t quite work as stated because one has to apply the closure property under union an uncountable number of times, but it turns out that if one works with the robust version of sumsets and uses a random sampling argument to approximate by the union of finitely many of the , then the argument can be made to work.

Using all of these closure properties, it turns out that one can start with an arbitrary critical pair and end up with a small set such that and are also critical pairs for all (say), where is the -fold sumset of . (Intuitively, if are thought of as secretly coming from the pullback of arcs by some character , then should be the pullback of a much shorter arc by the same character.) In particular, exhibits linear growth, in that for all . One can now use standard technology from inverse sumset theory to show first that has a very large Fourier coefficient (and thus is biased with respect to some character ), and secondly that is in fact almost of the form for some arc , from which it is not difficult to conclude similar statements for and and thus finish the proof of the inverse theorem.

In order to make the above argument rigorous, one has to be more precise about what the modifier “almost” means in the definition of a critical pair. I chose to do this in the language of “cheap” nonstandard analysis (aka asymptotic analysis), as discussed in this previous blog post; one could also have used the full-strength version of nonstandard analysis, but this does not seem to convey any substantial advantages. (One can also work in a more traditional “non-asymptotic” framework, but this requires one to keep much more careful account of various small error terms and leads to a messier argument.)

*[Update, Nov 15: Corrected the attribution of the inequality (1) to Kneser instead of Kemperman. Thanks to John Griesmer for pointing out the error.]*

A basic object of study in multiplicative number theory are the arithmetic functions: functions from the natural numbers to the complex numbers. Some fundamental examples of such functions include

- The constant function ;
- The Kronecker delta function ;
- The natural logarithm function ;
- The divisor function ;
- The von Mangoldt function , with defined to equal when is a power of a prime for some , and defined to equal zero otherwise; and
- The Möbius function , with defined to equal when is the product of distinct primes, and defined to equal zero otherwise.

Given an arithmetic function , we are often interested in statistics such as the summatory function

the logarithmically (or harmonically) weighted summatory function

or the Dirichlet series

In the latter case, one typically has to first restrict to those complex numbers whose real part is large enough in order to ensure the series on the right converges; but in many important cases, one can then extend the Dirichlet series to almost all of the complex plane by analytic continuation. One is also interested in correlations involving additive shifts, such as , but these are significantly more difficult to study and cannot be easily estimated by the methods of classical multiplicative number theory.

A key operation on arithmetic functions is that of Dirichlet convolution, which when given two arithmetic functions , forms a new arithmetic function , defined by the formula

Thus for instance , , , and for any arithmetic function . Dirichlet convolution and Dirichlet series are related by the fundamental formula

at least when the real part of is large enough that all sums involved become absolutely convergent (but in practice one can use analytic continuation to extend this identity to most of the complex plane). There is also the identity

at least when the real part of is large enough to justify interchange of differentiation and summation. As a consequence, many Dirichlet series can be expressed in terms of the Riemann zeta function , thus for instance

Much of the difficulty of multiplicative number theory can be traced back to the discrete nature of the natural numbers , which form a rather complicated abelian semigroup with respect to multiplication (in particular the set of generators is the set of prime numbers). One can obtain a simpler analogue of the subject by working instead with the half-infinite interval , which is a much simpler abelian semigroup under multiplication (being a one-dimensional Lie semigroup). (I will think of this as a sort of “completion” of at the infinite place , hence the terminology.) Accordingly, let us define a *continuous arithmetic function* to be a locally integrable function . The analogue of the summatory function (1) is then an integral

and similarly the analogue of (2) is

The analogue of the Dirichlet series is the Mellin-type transform

which will be well-defined at least if the real part of is large enough and if the continuous arithmetic function does not grow too quickly, and hopefully will also be defined elsewhere in the complex plane by analytic continuation.

For instance, the continuous analogue of the discrete constant function would be the constant function , which maps any to , and which we will denote by in order to keep it distinct from . The two functions and have approximately similar statistics; for instance one has

and

where is the harmonic number, and we are deliberately vague as to what the symbol means. Continuing this analogy, we would expect

which reflects the fact that has a simple pole at with residue , and no other poles. Note that the identity is initially only valid in the region , but clearly the right-hand side can be continued analytically to the entire complex plane except for the pole at , and so one can define in this region also.

In a similar vein, the logarithm function is approximately similar to the logarithm function , giving for instance the crude form

of Stirling’s formula, or the Dirichlet series approximation

The continuous analogue of Dirichlet convolution is multiplicative convolution using the multiplicative Haar measure : given two continuous arithmetic functions , one can define their convolution by the formula

Thus for instance . A short computation using Fubini’s theorem shows the analogue

of (3) whenever the real part of is large enough that Fubini’s theorem can be justified; similarly, differentiation under the integral sign shows that

again assuming that the real part of is large enough that differentiation under the integral sign (or some other tool like this, such as the Cauchy integral formula for derivatives) can be justified.

Direct calculation shows that for any complex number , one has

(at least for the real part of large enough), and hence by several applications of (5)

for any natural number . This can lead to the following heuristic: if a Dirichlet series behaves like a linear combination of poles , in that

for some set of poles and some coefficients and natural numbers (where we again are vague as to what means, and how to interpret the sum if the set of poles is infinite), then one should expect the arithmetic function to behave like the continuous arithmetic function

In particular, if we only have simple poles,

then we expect to have behave like continuous arithmetic function

Integrating this from to , this heuristically suggests an approximation

for the summatory function, and similarly

with the convention that is when , and similarly is when . One can make these sorts of approximations more rigorous by means of Perron’s formula (or one of its variants) combined with the residue theorem, provided that one has good enough control on the relevant Dirichlet series, but we will not pursue these rigorous calculations here. (But see for instance this previous blog post for some examples.)

For instance, using the more refined approximation

to the zeta function near , we have

we would expect that

and thus for instance

which matches what one actually gets from the Dirichlet hyperbola method (see e.g. equation (44) of this previous post).

Or, noting that has a simple pole at and assuming simple zeroes elsewhere, the log derivative will have simple poles of residue at and at all the zeroes, leading to the heuristic

suggesting that should behave like the continuous arithmetic function

leading for instance to the summatory approximation

which is a heuristic form of the Riemann-von Mangoldt explicit formula (see Exercise 45 of these notes for a rigorous version of this formula).

Exercise 1Go through some of the other explicit formulae listed at this Wikipedia page and give heuristic justifications for them (up to some lower order terms) by similar calculations to those given above.

Given the “adelic” perspective on number theory, I wonder if there are also -adic analogues of arithmetic functions to which a similar set of heuristics can be applied, perhaps to study sums such as . A key problem here is that there does not seem to be any good interpretation of the expression when is complex and is a -adic number, so it is not clear that one can analyse a Dirichlet series -adically. For similar reasons, we don’t have a canonical way to define for a Dirichlet character (unless its conductor happens to be a power of ), so there doesn’t seem to be much to say in the -aspect either.

Let be the Liouville function, thus is defined to equal when is the product of an even number of primes, and when is the product of an odd number of primes. The Chowla conjecture asserts that has the statistics of a random sign pattern, in the sense that

for all and all distinct natural numbers , where we use the averaging notation

For , this conjecture is equivalent to the prime number theorem (as discussed in this previous blog post), but the conjecture remains open for any .

In recent years, it has been realised that one can make more progress on this conjecture if one works instead with the logarithmically averaged version

of the conjecture, where we use the logarithmic averaging notation

Using the summation by parts (or telescoping series) identity

it is not difficult to show that the Chowla conjecture (1) for a given implies the logarithmically averaged conjecture (2). However, the converse implication is not at all clear. For instance, for , we have already mentioned that the Chowla conjecture

is equivalent to the prime number theorem; but the logarithmically averaged analogue

is significantly easier to show (a proof with the Liouville function replaced by the closely related Möbius function is given in this previous blog post). And indeed, significantly more is now known for the logarithmically averaged Chowla conjecture; in this paper of mine I had proven (2) for , and in this recent paper with Joni Teravainen, we proved the conjecture for all odd (with a different proof also given here).

In view of this emerging consensus that the logarithmically averaged Chowla conjecture was easier than the ordinary Chowla conjecture, it was thus somewhat of a surprise for me to read a recent paper of Gomilko, Kwietniak, and Lemanczyk who (among other things) established the following statement:

Theorem 1Assume that the logarithmically averaged Chowla conjecture (2) is true for all . Then there exists a sequence going to infinity such that the Chowla conjecture (1) is true for all along that sequence, that is to sayfor all and all distinct .

This implication does not use any special properties of the Liouville function (other than that they are bounded), and in fact proceeds by ergodic theoretic methods, focusing in particular on the ergodic decomposition of invariant measures of a shift into ergodic measures. Ergodic methods have proven remarkably fruitful in understanding these sorts of number theoretic and combinatorial problems, as could already be seen by the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recently by the work of Frantzikinakis and Host on Sarnak’s conjecture. (My first paper with Teravainen also uses ergodic theory tools.) Indeed, many other results in the subject were first discovered using ergodic theory methods.

On the other hand, many results in this subject that were first proven ergodic theoretically have since been reproven by more combinatorial means; my second paper with Teravainen is an instance of this. As it turns out, one can also prove Theorem 1 by a standard combinatorial (or probabilistic) technique known as the second moment method. In fact, one can prove slightly more:

Theorem 2Let be a natural number. Assume that the logarithmically averaged Chowla conjecture (2) is true for . Then there exists a set of natural numbers of logarithmic density (that is, ) such thatfor any distinct .

It is not difficult to deduce Theorem 1 from Theorem 2 using a diagonalisation argument. Unfortunately, the known cases of the logarithmically averaged Chowla conjecture ( and odd ) are currently insufficient to use Theorem 2 for any purpose other than to reprove what is already known to be true from the prime number theorem. (Indeed, the even cases of Chowla, in either logarithmically averaged or non-logarithmically averaged forms, seem to be far more powerful than the odd cases; see Remark 1.7 of this paper of myself and Teravainen for a related observation in this direction.)

We now sketch the proof of Theorem 2. For any distinct , we take a large number and consider the limiting the second moment

We can expand this as

If all the are distinct, the hypothesis (2) tells us that the inner averages goes to zero as . The remaining averages are , and there are of these averages. We conclude that

By Markov’s inequality (and (3)), we conclude that for any fixed , there exists a set of upper logarithmic density at least , thus

such that

By deleting at most finitely many elements, we may assume that consists only of elements of size at least (say).

For any , if we let be the union of for , then has logarithmic density . By a diagonalisation argument (using the fact that the set of tuples is countable), we can then find a set of natural numbers of logarithmic density , such that for every , every sufficiently large element of lies in . Thus for every sufficiently large in , one has

for some with . By Cauchy-Schwarz, this implies that

interchanging the sums and using and , this implies that

We conclude on taking to infinity that

as required.

## Recent Comments