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I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well ${\partial_{tt} u = -\nabla V(u)}$, here we try to embed dynamical systems into the incompressible Euler equations

$\displaystyle \partial_t u + \nabla_u u = - \mathrm{grad}_g p \ \ \ \ \ (1)$

$\displaystyle \mathrm{div}_g u = 0$

on a Riemannian manifold ${(M,g)}$. (One is particularly interested in the case of flat manifolds ${M}$, particularly ${{\bf R}^3}$ or ${({\bf R}/{\bf Z})^3}$, but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on ${M}$ (see this previous post for a discussion of this in the flat space case).

The Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field ${u}$. It is thus natural to compare the Euler equations with quadratic ODE of the form

$\displaystyle \partial_t y = B(y,y) \ \ \ \ \ (2)$

where ${y: {\bf R} \rightarrow {\bf R}^n}$ is the unknown solution, and ${B: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}^n}$ is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold ${(M,g)}$, which means that there is an injective linear map ${U: {\bf R}^n \rightarrow \Gamma(TM)}$ from ${{\bf R}^n}$ to smooth vector fields on ${M}$, as well as a bilinear map ${P: {\bf R}^n \times {\bf R}^n \rightarrow C^\infty(M)}$ to smooth scalar fields on ${M}$, such that the map ${y \mapsto (U(y), P(y,y))}$ takes solutions to (2) to solutions to (1), or equivalently that

$\displaystyle U(B(y,y)) + \nabla_{U(y)} U(y) = - \mathrm{grad}_g P(y,y)$

$\displaystyle \mathrm{div}_g U(y) = 0$

for all ${y \in {\bf R}^n}$.

For simplicity let us restrict ${M}$ to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form ${B}$ in (2) has to obey a cancellation condition

$\displaystyle \langle B(y,y), y \rangle = 0 \ \ \ \ \ (3)$

for some positive definite inner product ${\langle, \rangle: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}}$ on ${{\bf R}^n}$. The main result of the paper is the converse to this statement: if ${B}$ is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold ${(M,g)}$; the catch is that this manifold will depend on the form ${B}$ and on the dimension ${n}$ (in fact in the construction I have, ${M}$ is given explicitly as ${SO(n) \times ({\bf R}/{\bf Z})^{n+1}}$, with a funny metric on it that depends on ${B}$).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric ${g}$ by partially reformulating the Euler equations (1) in terms of the covelocity ${g \cdot u}$ (which is a ${1}$-form) instead of the velocity ${u}$. Using the freedom to modify the dimension of the underlying manifold ${M}$, one can also decouple the metric ${g}$ from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.

I am totally stunned to learn that Maryam Mirzakhani died today yesterday, aged 40, after a severe recurrence of the cancer she had been fighting for several years.  I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.

My first encounter with Maryam was in 2010, when I was giving some lectures at Stanford – one on Perelman’s proof of the Poincare conjecture, and another on random matrix theory.  I remember a young woman sitting in the front who asked perceptive questions at the end of both talks; it was only afterwards that I learned that it was Mirzakhani.  (I really wish I could remember exactly what the questions were, but I vaguely recall that she managed to put a nice dynamical systems interpretation on both of the topics of my talks.)

After she won the Fields medal in 2014 (as I posted about previously on this blog), we corresponded for a while.  The Fields medal is of course one of the highest honours one can receive in mathematics, and it clearly advances one’s career enormously; but it also comes with a huge initial burst of publicity, a marked increase in the number of responsibilities to the field one is requested to take on, and a strong expectation to serve as a public role model for mathematicians.  As the first female recipient of the medal, and also the first to come from Iran, Maryam was experiencing these pressures to a far greater extent than previous medallists, while also raising a small daughter and fighting off cancer.  I gave her what advice I could on these matters (mostly that it was acceptable – and in fact necessary – to say “no” to the vast majority of requests one receives).

Given all this, it is remarkable how productive she still was mathematically in the last few years.  Perhaps her greatest recent achievement has been her “magic wandtheorem with Alex Eskin, which is basically the analogue of the famous measure classification and orbit closure theorems of Marina Ratner, in the context of moduli spaces instead of unipotent flows on homogeneous spaces.  (I discussed Ratner’s theorems in this previous post.  By an unhappy coincidence, Ratner also passed away this month, aged 78.)  Ratner’s theorems are fundamentally important to any problem to which a homogeneous dynamical system can be associated (for instance, a special case of that theorem shows up in my work with Ben Green and Tamar Ziegler on the inverse conjecture for the Gowers norms, and on linear equations in primes), as it gives a good description of the equidistribution of any orbit of that system (if it is unipotently generated); and it seems the Eskin-Mirzakhani result will play a similar role in problems associated instead to moduli spaces.  The remarkable proof of this result – which now stands at over 200 pages, after three years of revision and updating – uses almost all of the latest techniques that had been developed for homogeneous dynamics, and ingeniously adapts them to the more difficult setting of moduli spaces, in a manner that had not been dreamed of being possible only a few years earlier.

Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers.  Today was a huge loss for Maryam’s family and friends, as well as for mathematics.

[EDIT, Jul 16: New York times obituary here.]

[EDIT, Jul 18: New Yorker memorial here.]

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential well equation

$\displaystyle \partial_{tt} u = - (\nabla F)(u) \ \ \ \ \ (1)$

for a particle ${u: {\bf R} \rightarrow {\bf R}^m}$ trapped in a potential well with potential ${F: {\bf R}^m \rightarrow {\bf R}}$, with ${F(z) \rightarrow +\infty}$ as ${z \rightarrow \infty}$. This ODE always admits global solutions from arbitrary initial positions ${u(0)}$ and initial velocities ${\partial_t u(0)}$, thanks to conservation of the Hamiltonian ${\frac{1}{2} |\partial_t u|^2 + F(u)}$. As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator ${\partial_{tt} u = - k^2 u}$ (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells universal in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let ${M}$ be a compact manifold, and let ${X}$ be a smooth vector field on ${M}$; to avoid degeneracies, let us take ${X}$ to be non-singular in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

$\displaystyle \partial_t u = X(u) \ \ \ \ \ (2)$

for ${u: {\bf R} \rightarrow M}$ are always global and almost periodic. Can we then find a (coercive) potential ${F: {\bf R}^m \rightarrow {\bf R}}$ for some ${m}$, as well as a smooth embedding ${\phi: M \rightarrow {\bf R}^m}$, such that every solution ${u}$ to (2) pushes forward under ${\phi}$ to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space ${{\bf R}^m \times {\bf R}^m}$, rather than position space ${{\bf R}^m}$, but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair ${(M,X)}$ as above, define a strongly adapted ${1}$-form to be a ${1}$-form ${\phi}$ on ${M}$ such that ${\phi(X)}$ is pointwise positive, and the Lie derivative ${{\mathcal L}_X \phi}$ is an exact ${1}$-form. We then have

Theorem 1 A smooth compact non-singular dynamics ${(M,X)}$ can be embedded smoothly in a potential well system if and only if it admits a strongly adapted ${1}$-form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space ${{\bf R}^m \times {\bf R}^m}$) admit a strongly adapted ${1}$-form, namely the canonical ${1}$-form ${p dq}$, whose Lie derivative is the derivative ${dL}$ of the Lagrangian ${L := \frac{1}{2} |\partial_t u|^2 - F(u)}$ and is thus exact. The converse “if” direction is mainly a consequence of the Nash embedding theorem, and follows the arguments used in my previous paper.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than ${{\bf R}^m}$, or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics ${(M,X)}$ that do not support strongly adapted ${1}$-forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field ${(\sin(2\pi x), \cos(2\pi x))}$ on the ${2}$-torus ${({\bf R}/{\bf Z})^2}$ had no strongly adapted ${1}$-forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted ${1}$-form (the derivative ${dt}$ of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)

A few days ago, I was talking with Ed Dunne, who is currently the Executive Editor of Mathematical Reviews (and in particular with its online incarnation at MathSciNet).  At the time, I was mentioning how laborious it was for me to create a BibTeX file for dozens of references by using MathSciNet to locate each reference separately, and to export each one to BibTeX format.  He then informed me that underneath to every MathSciNet reference there was a little link to add the reference to a Clipboard, and then one could export the entire Clipboard at once to whatever format one wished.  In retrospect, this was a functionality of the site that had always been visible, but I had never bothered to explore it, and now I can populate a BibTeX file much more quickly.

This made me realise that perhaps there are many other useful features of popular mathematical tools out there that only a few users actually know about, so I wanted to create a blog post to encourage readers to post their own favorite tools, or features of tools, that are out there, often in plain sight, but not always widely known.  Here are a few that I was able to recall from my own workflow (though for some of them it took quite a while to consciously remember, since I have been so used to them for so long!):

1. TeX for Gmail.  A Chrome plugin that lets one write TeX symbols in emails sent through Gmail (by writing the LaTeX code and pressing a hotkey, usually F8).
2. Boomerang for Gmail.  Another Chrome plugin for Gmail, which does two main things.  Firstly, it can “boomerang” away an email from your inbox to return at some specified later date (e.g. one week from today).  I found this useful to declutter my inbox regarding mail that I needed to act on in the future, but was unable to deal with at present due to travel, or because I was waiting for some other piece of data to arrive first.   Secondly, it can send out email with some specified delay (e.g. by tomorrow morning), giving one time to cancel the email if necessary.  (Thanks to Julia Wolf for telling me about Boomerang!)
3. Which just reminds me, the Undo Send feature on Gmail has saved me from embarrassment a few times (but one has to set it up first; it delays one’s emails by a short period, such as 30 seconds, during which time it is possible to undo the email).
4. LaTeX rendering in Inkscape.  I used to use plain text to write mathematical formulae in my images, which always looked terrible.  It took me years to realise that Inkscape had the functionality to compile LaTeX within it.
5. Bookmarks in TeXnicCenter.  I probably only use a tiny fraction of the functionality that TeXnicCenter offers, but one little feature I quite like is the ability to bookmark a portion of the TeX file (e.g. the bibliography at the end, or the place one is currently editing) with one hot-key (Ctrl-F2) and then one can cycle quickly between one bookmarked location and another with some further hot-keys (F2 and shift-F2).
6. Actually, there are a number of Windows keyboard shortcuts that are worth experimenting with (and similarly for Mac or Linux systems of course).
7. Detexify has been the quickest way for me to locate the TeX code for a symbol that I couldn’t quite remember (or when hunting for a new symbol that would roughly be shaped like something I had in mind).
8. For writing LaTeX on my blog, I use Luca Trevisan’s LaTeX to WordPress Python script (together with a little batch file I wrote to actually run the python script).
9. Using the camera on my phone to record a blackboard computation or a slide (or the wifi password at a conference centre, or any other piece of information that is written or displayed really).  If the phone is set up properly this can be far quicker than writing it down with pen and paper.  (I guess this particular trick is now quite widely used, but I still see people surprised when someone else uses a phone instead of a pen to record things.)
10. Using my online calendar not only to record scheduled future appointments, but also to block out time to do specific tasks (e.g. reserve 2-3pm at Tuesday to read paper X, or do errand Y).  I have found I am able to get a much larger fraction of my “to do” list done on days in which I had previously blocked out such specific chunks of time, as opposed to days in which I had left several hours unscheduled (though sometimes those hours were also very useful for finding surprising new things to do that I had not anticipated).  (I learned of this little trick online somewhere, but I have long since lost the original reference.)

Anyway, I would very much like to hear what other little tools or features other readers have found useful in their work.

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges“, submitted to Proceedings of the London Mathematical Society. This paper is concerned with the estimation of correlations such as

$\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ \ \ \ \ (1)$

for medium-sized ${h}$ and large ${X}$, where ${\Lambda}$ is the von Mangoldt function; we also consider variants of this sum in which one of the von Mangoldt functions is replaced with a (higher order) divisor function, but for sake of discussion let us focus just on the sum (1). Understanding this sum is very closely related to the problem of finding pairs of primes that differ by ${h}$; for instance, if one could establish a lower bound

$\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+2) \gg X$

then this would easily imply the twin prime conjecture.

The (first) Hardy-Littlewood conjecture asserts an asymptotic

$\displaystyle \sum_{n \leq X} \Lambda(n) \Lambda(n+h) = {\mathfrak S}(h) X + o(X) \ \ \ \ \ (2)$

as ${X \rightarrow \infty}$ for any fixed positive ${h}$, where the singular series ${{\mathfrak S}(h)}$ is an arithmetic factor arising from the irregularity of distribution of ${\Lambda}$ at small moduli, defined explicitly by

$\displaystyle {\mathfrak S}(h) := 2 \Pi_2 \prod_{p|h; p>2} \frac{p-2}{p-1}$

when ${h}$ is even, and ${{\mathfrak S}(h)=0}$ when ${h}$ is odd, where

$\displaystyle \Pi_2 := \prod_{p>2} (1-\frac{1}{(p-1)^2}) = 0.66016\dots$

is (half of) the twin prime constant. See for instance this previous blog post for a a heuristic explanation of this conjecture. From the previous discussion we see that (2) for ${h=2}$ would imply the twin prime conjecture. Sieve theoretic methods are only able to provide an upper bound of the form ${ \sum_{n \leq X} \Lambda(n) \Lambda(n+h) \ll {\mathfrak S}(h) X}$.

Needless to say, apart from the trivial case of odd ${h}$, there are no values of ${h}$ for which the Hardy-Littlewood conjecture is known. However there are some results that say that this conjecture holds “on the average”: in particular, if ${H}$ is a quantity depending on ${X}$ that is somewhat large, there are results that show that (2) holds for most (i.e. for ${1-o(1)}$) of the ${h}$ betwen ${0}$ and ${H}$. Ideally one would like to get ${H}$ as small as possible, in particular one can view the full Hardy-Littlewood conjecture as the endpoint case when ${H}$ is bounded.

The first results in this direction were by van der Corput and by Lavrik, who established such a result with ${H = X}$ (with a subsequent refinement by Balog); Wolke lowered ${H}$ to ${X^{5/8+\varepsilon}}$, and Mikawa lowered ${H}$ further to ${X^{1/3+\varepsilon}}$. The main result of this paper is a further lowering of ${H}$ to ${X^{8/33+\varepsilon}}$. In fact (as in the preceding works) we get a better error term than ${o(X)}$, namely an error of the shape ${O_A( X \log^{-A} X)}$ for any ${A}$.

Our arguments initially proceed along standard lines. One can use the Hardy-Littlewood circle method to express the correlation in (2) as an integral involving exponential sums ${S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}$. The contribution of “major arc” ${\alpha}$ is known by a standard computation to recover the main term ${{\mathfrak S}(h) X}$ plus acceptable errors, so it is a matter of controlling the “minor arcs”. After averaging in ${h}$ and using the Plancherel identity, one is basically faced with establishing a bound of the form

$\displaystyle \int_{\beta-1/H}^{\beta+1/H} |S(\alpha)|^2\ d\alpha \ll_A X \log^{-A} X$

for any “minor arc” ${\beta}$. If ${\beta}$ is somewhat close to a low height rational ${a/q}$ (specifically, if it is within ${X^{-1/6-\varepsilon}}$ of such a rational with ${q = O(\log^{O(1)} X)}$), then this type of estimate is roughly of comparable strength (by another application of Plancherel) to the best available prime number theorem in short intervals on the average, namely that the prime number theorem holds for most intervals of the form ${[x, x + x^{1/6+\varepsilon}]}$, and we can handle this case using standard mean value theorems for Dirichlet series. So we can restrict attention to the “strongly minor arc” case where ${\beta}$ is far from such rationals.

The next step (following some ideas we found in a paper of Zhan) is to rewrite this estimate not in terms of the exponential sums ${S(\alpha) := \sum_{n \leq X} \Lambda(n) e(\alpha n)}$, but rather in terms of the Dirichlet polynomial ${F(s) := \sum_{n \sim X} \frac{\Lambda(n)}{n^s}}$. After a certain amount of computation (including some oscillatory integral estimates arising from stationary phase), one is eventually reduced to the task of establishing an estimate of the form

$\displaystyle \int_{t \sim \lambda X} (\sum_{t-\lambda H}^{t+\lambda H} |F(\frac{1}{2}+it')|\ dt')^2\ dt \ll_A \lambda^2 H^2 X \log^{-A} X$

for any ${X^{-1/6-\varepsilon} \ll \lambda \ll \log^{-B} X}$ (with ${B}$ sufficiently large depending on ${A}$).

The next step, which is again standard, is the use of the Heath-Brown identity (as discussed for instance in this previous blog post) to split up ${\Lambda}$ into a number of components that have a Dirichlet convolution structure. Because the exponent ${8/33}$ we are shooting for is less than ${1/4}$, we end up with five types of components that arise, which we call “Type ${d_1}$“, “Type ${d_2}$“, “Type ${d_3}$“, “Type ${d_4}$“, and “Type II”. The “Type II” sums are Dirichlet convolutions involving a factor supported on a range ${[X^\varepsilon, X^{-\varepsilon} H]}$ and is quite easy to deal with; the “Type ${d_j}$” terms are Dirichlet convolutions that resemble (non-degenerate portions of) the ${j^{th}}$ divisor function, formed from convolving together ${j}$ portions of ${1}$. The “Type ${d_1}$” and “Type ${d_2}$” terms can be estimated satisfactorily by standard moment estimates for Dirichlet polynomials; this already recovers the result of Mikawa (and our argument is in fact slightly more elementary in that no Kloosterman sum estimates are required). It is the treatment of the “Type ${d_3}$” and “Type ${d_4}$” sums that require some new analysis, with the Type ${d_3}$ terms turning to be the most delicate. After using an existing moment estimate of Jutila for Dirichlet L-functions, matters reduce to obtaining a family of estimates, a typical one of which (relating to the more difficult Type ${d_3}$ sums) is of the form

$\displaystyle \int_{t - H}^{t+H} |M( \frac{1}{2} + it')|^2\ dt' \ll X^{\varepsilon^2} H \ \ \ \ \ (3)$

for “typical” ordinates ${t}$ of size ${X}$, where ${M}$ is the Dirichlet polynomial ${M(s) := \sum_{n \sim X^{1/3}} \frac{1}{n^s}}$ (a fragment of the Riemann zeta function). The precise definition of “typical” is a little technical (because of the complicated nature of Jutila’s estimate) and will not be detailed here. Such a claim would follow easily from the Lindelof hypothesis (which would imply that ${M(1/2 + it) \ll X^{o(1)}}$) but of course we would like to have an unconditional result.

At this point, having exhausted all the Dirichlet polynomial estimates that are usefully available, we return to “physical space”. Using some further Fourier-analytic and oscillatory integral computations, we can estimate the left-hand side of (3) by an expression that is roughly of the shape

$\displaystyle \frac{H}{X^{1/3}} \sum_{\ell \sim X^{1/3}/H} |\sum_{m \sim X^{1/3}} e( \frac{t}{2\pi} \log \frac{m+\ell}{m-\ell} )|.$

The phase ${\frac{t}{2\pi} \log \frac{m+\ell}{m-\ell}}$ can be Taylor expanded as the sum of ${\frac{t_j \ell}{\pi m}}$ and a lower order term ${\frac{t_j \ell^3}{3\pi m^3}}$, plus negligible errors. If we could discard the lower order term then we would get quite a good bound using the exponential sum estimates of Robert and Sargos, which control averages of exponential sums with purely monomial phases, with the averaging allowing us to exploit the hypothesis that ${t}$ is “typical”. Figuring out how to get rid of this lower order term caused some inefficiency in our arguments; the best we could do (after much experimentation) was to use Fourier analysis to shorten the sums, estimate a one-parameter average exponential sum with a binomial phase by a two-parameter average with a monomial phase, and then use the van der Corput ${B}$ process followed by the estimates of Robert and Sargos. This rather complicated procedure works up to ${H = X^{8/33+\varepsilon}}$ it may be possible that some alternate way to proceed here could improve the exponent somewhat.

In a sequel to this paper, we will use a somewhat different method to reduce ${H}$ to a much smaller value of ${\log^{O(1)} X}$, but only if we replace the correlations ${\sum_{n \leq X} \Lambda(n) \Lambda(n+h)}$ by either ${\sum_{n \leq X} \Lambda(n) d_k(n+h)}$ or ${\sum_{n \leq X} d_k(n) d_l(n+h)}$, and also we now only save a ${o(1)}$ in the error term rather than ${O_A(\log^{-A} X)}$.