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This is the second “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this previous thread. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

We now have the following proposition (see this page for a proof sketch) that looks like it can give a numerically feasible approach to bound :

Proposition 1Suppose that one has parameters obeying the following properties:

- All the zeroes of with are real.
- There are no zeroes with in the region .
- There are no zeroes with and .
Then one has .

The first hypothesis is already known for up to about (we should find out exactly what we can reach here). Preliminary calculations suggest that we can obtain the third item provided that . The second hypothesis requires good numerical calculation for , to which we now turn.

The initial definition of is given by the formula

where

This formula has proven numerically computable to acceptable error up until about the first hundred zeroes of , but degrades after that, and so other exact or approximate formulae for are needed. One possible exact formula that could be useful is

where

and

and can be chosen arbitrarily. We are still trying to see if this can be implemented numerically to give better accuracy than the previous formula.

It seems particularly promising to develop a generalisation of the Riemann-Siegel approximate functional equation for . Preliminary computations suggest in particular that we have the approximation

where

Some very preliminary numerics suggest that this formula is reasonably accurate even for moderate values of , though further numerical verification is needed. As a proof of concept, one could take this approximation as exact for the purposes of seeing what ranges of one can feasibly compute with (and for extremely large values of , we will presumably have to introduce some version of the Odlyzko-Schönhage algorithm. Of course, to obtain a rigorous result, we will eventually need a rigorous version of this formula with explicit error bounds. It may also be necessary to add more terms to the approximation to reduce the size of the error.

Sujit Nair has kindly summarised for me the current state of affairs with the numerics as follows:

—

- We need a real milestone and work backward to set up intermediate goals. This will definitely help bring in focus!
- So far, we have some utilities to compute zeroes of with a nonlinear solver which uses roots of as an initial condition. The solver is a wrapper around MINPACK’s implementation of Powell’s method. There is some room for optimization. For example, we aren’t providing the solver with the analytical Jacobian which speeds up the computation and increases accuracy.
- We have some results in the output folder which contains the first 1000 roots of for some small values of , etc. They need some more organization and visualization.

We have a decent initial start but we have some ways to go. Moving forward, here is my proposition for some areas of focus. We should expand and prioritize after some open discussion.

- Short term Optimize the existing framework and target to have the first million zeros of (for a reasonable range of ) and the corresponding plots. With better engineering practice and discipline, I am confident we can get to a few tens of millions range. Some things which will help include parallelization, iterative approaches (using zeroes of to compute zeroes of ), etc.
- Medium term We need to explore better ways to represent the zeros and compute them. An analogy is the computation of Riemann zeroes up to height . It is computed by computing the sign changes of (page 119 of Edwards) and by exploiting the speed up with the Riemann-Siegel formulation (over Euler-Maclaurin). For larger values of , I am not sure the root solver based approach is going to work to understand the gaps between zeroes.
- Long term We also need a better understanding of the errors involved in the computation — truncation, hardware/software, etc.

This is the first official “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant . Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

The proposal naturally splits into at least three separate (but loosely related) topics:

- Numerical computation of the entire functions , with the ultimate aim of establishing zero-free regions of the form for various .
- Improved understanding of the dynamics of the zeroes of .
- Establishing the zero-free nature of when and is sufficiently large depending on and .

Below the fold, I will present each of these topics in turn, to initiate further discussion in each of them. (I thought about splitting this post into three to have three separate discussions, but given the current volume of comments, I think we should be able to manage for now having all the comments in a single post. If this changes then of course we can split up some of the discussion later.)

To begin with, let me present some formulae for computing (inspired by similar computations in the Ki-Kim-Lee paper) which may be useful. The initial definition of is

where

is a variant of the Jacobi theta function. We observe that in fact extends analytically to the strip

as has positive real part on this strip. One can use the Poisson summation formula to verify that is even, (see this previous post for details). This lets us obtain a number of other formulae for . Most obviously, one can unfold the integral to obtain

In my previous paper with Brad, we used this representation, combined with Fubini’s theorem to swap the sum and integral, to obtain a useful series representation for in the case. Unfortunately this is not possible in the case because expressions such as diverge as approaches . Nevertheless we can still perform the following contour integration manipulation. Let be fixed. The function decays super-exponentially fast (much faster than , in particular) as with ; as is even, we also have this decay as with (this is despite each of the summands in having much slower decay in this direction – there is considerable cancellation!). Hence by the Cauchy integral formula we have

Splitting the horizontal line from to at and using the even nature of , we thus have

Using the functional equation , we thus have the representation

where is the oscillatory integral

The formula (2) is valid for any . Naively one would think that it would be simplest to take ; however, when and is large (with bounded), it seems asymptotically better to take closer to , in particular something like seems to be a reasonably good choice. This is because the integrand in (3) becomes significantly less oscillatory and also much lower in amplitude; the term in (3) now generates a factor roughly comparable to (which, as we will see below, is the main term in the decay asymptotics for ), while the term still exhibits a reasonable amount of decay as . We will use the representation (2) in the asymptotic analysis of below, but it may also be a useful representation to use for numerical purposes.

Building on the interest expressed in the comments to this previous post, I am now formally proposing to initiate a “Polymath project” on the topic of obtaining new upper bounds on the de Bruijn-Newman constant . The purpose of this post is to describe the proposal and discuss the scope and parameters of the project.

De Bruijn introduced a family of entire functions for each real number , defined by the formula

where is the super-exponentially decaying function

As discussed in this previous post, the Riemann hypothesis is equivalent to the assertion that all the zeroes of are real.

De Bruijn and Newman showed that there existed a real constant – the de Bruijn-Newman constant – such that has all zeroes real whenever , and at least one non-real zero when . In particular, the Riemann hypothesis is equivalent to the upper bound . In the opposite direction, several lower bounds on have been obtained over the years, most recently in my paper with Brad Rodgers where we showed that , a conjecture of Newman.

As for upper bounds, de Bruijn showed back in 1950 that . The only progress since then has been the work of Ki, Kim and Lee in 2009, who improved this slightly to . The primary proposed aim of this Polymath project is to obtain further explicit improvements to the upper bound of . Of course, if we could lower the upper bound all the way to zero, this would solve the Riemann hypothesis, but I do not view this as a realistic outcome of this project; rather, the upper bounds that one could plausibly obtain by known methods and numerics would be comparable in achievement to the various numerical verifications of the Riemann hypothesis that exist in the literature (e.g., that the first non-trivial zeroes of the zeta function lie on the critical line, for various large explicit values of ).

In addition to the primary goal, one could envisage some related secondary goals of the project, such as a better understanding (both analytic and numerical) of the functions (or of similar functions), and of the dynamics of the zeroes of these functions. Perhaps further potential goals could emerge in the discussion to this post.

I think there is a plausible plan of attack on this project that proceeds as follows. Firstly, there are results going back to the original work of de Bruijn that demonstrate that the zeroes of become attracted to the real line as increases; in particular, if one defines to be the supremum of the imaginary parts of all the zeroes of , then it is known that this quantity obeys the differential inequality

whenever is positive; furthermore, once for some , then for all . I hope to explain this in a future post (it is basically due to the attraction that a zero off the real axis has to its complex conjugate). As a corollary of this inequality, we have the upper bound

for any real number . For instance, because all the non-trivial zeroes of the Riemann zeta function lie in the critical strip , one has , which when inserted into (2) gives . The inequality (1) also gives for all . If we could find some explicit between and where we can improve this upper bound on by an explicit constant, this would lead to a new upper bound on .

Secondly, the work of Ki, Kim and Lee (based on an analysis of the various terms appearing in the expression for ) shows that for any positive , all but finitely many of the zeroes of are real (in contrast with the situation, where it is still an open question as to whether the proportion of non-trivial zeroes of the zeta function on the critical line is asymptotically equal to ). As a key step in this analysis, Ki, Kim, and Lee show that for any and , there exists a such that all the zeroes of with real part at least , have imaginary part at most . Ki, Kim and Lee do not explicitly compute how depends on and , but it looks like this bound could be made effective.

If so, this suggests a possible strategy to get a new upper bound on :

- Select a good choice of parameters .
- By refining the Ki-Kim-Lee analysis, find an explicit such that all zeroes of with real part at least have imaginary part at most .
- By a numerical computation (e.g. using the argument principle), also verify that zeroes of with real part between and have imaginary part at most .
- Combining these facts, we obtain that ; hopefully, one can insert this into (2) and get a new upper bound for .

Of course, there may also be alternate strategies to upper bound , and I would imagine this would also be a legitimate topic of discussion for this project.

One appealing thing about the above strategy for the purposes of a polymath project is that it naturally splits the project into several interacting but reasonably independent parts: an analytic part in which one tries to refine the Ki-Kim-Lee analysis (based on explicitly upper and lower bounding various terms in a certain series expansion for – I may detail this later in a subsequent post); a numerical part in which one controls the zeroes of in a certain finite range; and perhaps also a dynamical part where one sees if there is any way to improve the inequality (2). For instance, the numerical “team” might, over time, be able to produce zero-free regions for with an increasingly large value of , while in parallel the analytic “team” might produce increasingly smaller values of beyond which they can control zeroes, and eventually the two bounds would meet up and we obtain a new bound on . This factoring of the problem into smaller parts was also a feature of the successful Polymath8 project on bounded gaps between primes.

The project also resembles Polymath8 in another aspect: that there is an obvious way to numerically measure progress, by seeing how the upper bound for decreases over time (and presumably there will also be another metric of progress regarding how well we can control in terms of and ). However, in Polymath8 the final measure of progress (the upper bound on gaps between primes) was a natural number, and thus could not decrease indefinitely. Here, the bound will be a real number, and there is a possibility that one may end up having an infinite descent in which progress slows down over time, with refinements to increasingly less significant digits of the bound as the project progresses. Because of this, I think it makes sense to follow recent Polymath projects and place an expiration date for the project, for instance one year after the launch date, in which we will agree to end the project and (if the project was successful enough) write up the results, unless there is consensus at that time to extend the project. (In retrospect, we should probably have imposed similar sunset dates on older Polymath projects, some of which have now been inactive for years, but that is perhaps a discussion for another time.)

Some Polymath projects have been known for a breakneck pace, making it hard for some participants to keep up. It’s hard to control these things, but I am envisaging a relatively leisurely project here, perhaps taking the full year mentioned above. It may well be that as the project matures we will largely be waiting for the results of lengthy numerical calculations to come in, for instance. Of course, as with previous projects, we would maintain some wiki pages (and possibly some other resources, such as a code repository) to keep track of progress and also to summarise what we have learned so far. For instance, as was done with some previous Polymath projects, we could begin with some “online reading seminars” where we go through some relevant piece of literature (most obviously the Ki-Kim-Lee paper, but there may be other resources that become relevant, e.g. one could imagine the literature on numerical verification of RH to be of value).

One could also imagine some incidental outcomes of this project, such as a more efficient way to numerically establish zero free regions for various analytic functions of interest; in particular, the project may well end up focusing on some other aspect of mathematics than the specific questions posed here.

Anyway, I would be interested to hear in the comments below from others who might be interested in participating, or at least observing, this project, particularly if they have suggestions regarding the scope and direction of the project, and on organisational structure (e.g. if one should start with reading seminars, or some initial numerical exploration of the functions , etc..) One could also begin some preliminary discussion of the actual mathematics of the project itself, though (in line with the leisurely pace I was hoping for), I expect that the main burst of mathematical activity would happen later, once the project is formally launched (with wiki page resources, blog posts dedicated to specific aspects of the project, etc.).

In this post we assume the Riemann hypothesis and the simplicity of zeroes, thus the zeroes of in the critical strip take the form for some real number ordinates . From the Riemann-von Mangoldt formula, one has the asymptotic

as ; in particular, the spacing should behave like on the average. However, it can happen that some gaps are unusually small compared to other nearby gaps. For the sake of concreteness, let us define a Lehmer pair to be a pair of adjacent ordinates such that

The specific value of constant is not particularly important here; anything larger than would suffice. An example of such a pair would be the classical pair

discovered by Lehmer. It follows easily from the main results of Csordas, Smith, and Varga that if an infinite number of Lehmer pairs (in the above sense) existed, then the de Bruijn-Newman constant is non-negative. This implication is now redundant in view of the unconditional results of this recent paper of Rodgers and myself; however, the question of whether an infinite number of Lehmer pairs exist remain open.

In this post, I sketch an argument that Brad and I came up with (as initially suggested by Odlyzko) the GUE hypothesis implies the existence of infinitely many Lehmer pairs. We argue probabilistically: pick a sufficiently large number , pick at random from to (so that the average gap size is close to ), and prove that the Lehmer pair condition (1) occurs with positive probability.

Introduce the renormalised ordinates for , and let be a small absolute constant (independent of ). It will then suffice to show that

(say) with probability , since the contribution of those outside of can be absorbed by the factor with probability .

As one consequence of the GUE hypothesis, we have with probability . Thus, if , then has density . Applying the Hardy-Littlewood maximal inequality, we see that with probability , we have

which implies in particular that

for all . This implies in particular that

and so it will suffice to show that

(say) with probability .

By the GUE hypothesis (and the fact that is independent of ), it suffices to show that a Dyson sine process , normalised so that is the first positive point in the process, obeys the inequality

with probability . However, if we let be a moderately large constant (and assume small depending on ), one can show using -point correlation functions for the Dyson sine process (and the fact that the Dyson kernel equals to second order at the origin) that

for any natural number , where denotes the number of elements of the process in . For instance, the expression can be written in terms of the three-point correlation function as

which can easily be estimated to be (since in this region), and similarly for the other estimates claimed above.

Since for natural numbers , the quantity is only positive when , we see from the first three estimates that the event that occurs with probability . In particular, by Markov’s inequality we have the conditional probabilities

and thus, if is large enough, and small enough, it will be true with probability that

and

and simultaneously that

for all natural numbers . This implies in particular that

and

for all , which gives (2) for small enough.

Remark 1The above argument needed the GUE hypothesis for correlations up to fourth order (in order to establish (3)). It might be possible to reduce the number of correlations needed, but I do not see how to obtain the claim just using pair correlations only.

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.

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).

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.

Joni Teräväinen and I have just uploaded to the arXiv our paper “Odd order cases of the logarithmically averaged Chowla conjecture“, submitted to J. Numb. Thy. Bordeaux. This paper gives an alternate route to one of the main results of our previous paper, and more specifically reproves the asymptotic

for all odd and all integers (that is to say, all the odd order cases of the logarithmically averaged Chowla conjecture). Our previous argument relies heavily on some deep ergodic theory results of Bergelson-Host-Kra, Leibman, and Le (and was applicable to more general multiplicative functions than the Liouville function ); here we give a shorter proof that avoids ergodic theory (but instead requires the Gowers uniformity of the (W-tricked) von Mangoldt function, established in several papers of Ben Green, Tamar Ziegler, and myself). The proof follows the lines sketched in the previous blog post. In principle, due to the avoidance of ergodic theory, the arguments here have a greater chance to be made quantitative; however, at present the known bounds on the Gowers uniformity of the von Mangoldt function are qualitative, except at the level, which is unfortunate since the first non-trivial odd case requires quantitative control on the level. (But it may be possible to make the Gowers uniformity bounds for quantitative if one assumes GRH, although when one puts everything together, the actual decay rate obtained in (1) is likely to be poor.)

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that

whenever were sequences going to infinity, were distinct integers, and were -bounded multiplicative functions which were *non-pretentious* in the sense that

for all Dirichlet characters and for . Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

for fixed any non-zero , where was the Liouville function.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

for all odd and all integers (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument ).

For the more general Elliott conjecture, we can show that

for any , any integers and any bounded multiplicative functions , unless the product *weakly pretends to be a Dirichlet character * in the sense that

This can be seen to imply (2) as a special case. Even when *does* pretend to be a Dirichlet character , we can still say something: if the limits

exist for each (which can be guaranteed if we pass to a suitable subsequence), then is the uniform limit of periodic functions , each of which is –isotypic in the sense that whenever are integers with coprime to the periods of and . This does not pin down the value of any single correlation , but does put significant constraints on how these correlations may vary with .

Among other things, this allows us to show that all possible length four sign patterns of the Liouville function occur with positive density, and all possible length four sign patterns occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function . The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime and observe that for any , which allows us to rewrite (3) as

Making the change of variables , we obtain

The difference between and is negligible in the limit (here is where we crucially rely on the log-averaging), hence

and thus by (3) we have

The entropy decrement argument can be used to show that the latter limit is small for most (roughly speaking, this is because the factors behave like independent random variables as varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the factors). We thus obtain the approximate isotopy property

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express as a multiple correlation

for some probability space equipped with a measure-preserving invertible map . Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

where is a nilsequence, and goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on , which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error , we can now combine (5) to conclude that

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up further into a periodic piece and an “irrational” or “minor arc” piece . The contribution of the minor arc piece can be shown to mostly cancel itself out after dilating by primes and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

which already shows (heuristically, at least) the claim that can be approximated by periodic functions which are isotopic in the sense that

But if is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes that are modulo the period of , and conclude now that vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in using the “-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

where ranges over a large range of integers coprime to some primorial . On the other hand, by iterating (4) we have

for most semiprimes , and by again averaging over semiprimes one can obtain an approximation of the form

For odd, one can combine the two approximations to conclude that . (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

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