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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

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

for medium-sized ${h}$ and large ${X}$, where ${k \geq l \geq 1}$ are natural numbers and ${d_k(n) = \sum_{n = m_1 \dots m_k} 1}$ is the ${k^{th}}$ divisor function (actually our methods can also treat a generalisation in which ${k}$ 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

$\displaystyle \sum_{n \leq X} d_k(n) d_l(n+h) = P_{k,l,h}( \log X ) X + O( X^{1/2+\varepsilon})$

for any fixed ${\varepsilon>0}$, where ${P_{k,l,h}}$ is a certain explicit (but rather complicated) polynomial of degree ${k+l-1}$. Such asymptotics are known when ${l \leq 2}$, but remain open for ${k \geq l \geq 3}$. In the previous paper, we were able to obtain a weaker bound of the form

$\displaystyle \sum_{n \leq X} d_k(n) d_l(n+h) = P_{k,l,h}( \log X ) X + O_A( X \log^{-A} X)$

for ${1-O_A(\log^{-A} X)}$ of the shifts ${-H \leq h \leq H}$, whenever the shift range ${H}$ lies between ${X^{8/33+\varepsilon}}$ and ${X^{1-\varepsilon}}$. But the methods become increasingly hard to use as ${H}$ gets smaller. In this paper, we use a rather different method to obtain the even weaker bound

$\displaystyle \sum_{n \leq X} d_k(n) d_l(n+h) = (1+o(1)) P_{k,l,h}( \log X ) X$

for ${1-o(1)}$ of the shifts ${-H \leq h \leq H}$, where ${H}$ can now be as short as ${H = \log^{10^4 k \log k} X}$. The constant ${10^4}$ can be improved, but there are serious obstacles to using our method to go below ${\log^{k \log k} X}$ (as the exceptionally large values of ${d_k}$ 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 ${d_k,d_l}$ 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

$\displaystyle S(\alpha) := \sum_{n \leq X} d_k(n) e(\alpha).$

Actually, it is convenient to first prune ${d_k}$ slightly by zeroing out this function on “atypical” numbers ${n}$ 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 ${S(\alpha)}$ for “major arc” ${\alpha}$ can be treated by standard techniques (and is the source of the main term ${P_{k,l,h}(\log X) X}$; the main difficulty comes from treating the contribution of “minor arc” ${\alpha}$.

In our previous paper on bounded multiplicative functions, we used Plancherel’s theorem to estimate the global ${L^2}$ norm ${\int_{{\bf R}/{\bf Z}} |S(\alpha)|^2\ d\alpha}$, and then also used the Katai-Bourgain-Sarnak-Ziegler orthogonality criterion to control local ${L^2}$ norms ${\int_I |S(\alpha)|^2\ d\alpha}$, where ${I}$ was a minor arc interval of length about ${1/H}$, and these two estimates together were sufficient to get a good bound on correlations by an application of Hölder’s inequality. For ${d_k}$, it is more convenient to use Dirichlet series methods (and Ramaré-type factorisations of such Dirichlet series) to control local ${L^2}$ 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 ${d_k}$, 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 ${L^2}$ bound will definitely be unusable, because the ${\ell^2}$ sum ${\sum_{n \leq X} d_k(n)^2}$ has too many unwanted factors of ${\log X}$. Fortunately, we can substitute this global ${L^2}$ bound with a “large values” bound that controls expressions such as

$\displaystyle \sum_{i=1}^J \int_{I_i} |S(\alpha)|^2\ d\alpha$

for a moderate number of disjoint intervals ${I_1,\dots,I_J}$, with a bound that is slightly better (for ${J}$ a medium-sized power of ${\log X}$) than what one would have obtained by bounding each integral ${\int_{I_i} |S(\alpha)|^2\ d\alpha}$ separately. (One needs to save more than ${J^{1/2}}$ for the argument to work; we end up saving a factor of about ${J^{3/4}}$.) 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

$\displaystyle \sum_{i=1}^J (\int_X^{2X} |\sum_{x \leq n \leq x+H} d_k(n) e(\alpha_i n)|^2\ dx)^{1/2},$

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

$\displaystyle |\sum_{x \leq n \leq x+H} \tilde d_k(n) e((\alpha_i -\alpha_{i'}) n)|$

after various averaging in the ${x, i,i'}$ parameters. These local Fourier coefficients of ${\tilde d_k}$ turn out to be small on average unless ${\alpha_i -\alpha_{i'}}$ 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 ${\alpha_i}$ 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 ${\alpha_i}$ 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 1 Let ${G = (G,\cdot)}$ be a group. Suppose one has a “seminorm” function ${\| \|: G \rightarrow [0,+\infty)}$ which obeys the triangle inequality

$\displaystyle \|xy \| \leq \|x\| + \|y\|$

for all ${x,y \in G}$, with equality whenever ${x=y}$. Then the seminorm factors through the abelianisation map ${G \mapsto G/[G,G]}$.

Proof: By the triangle inequality, it suffices to show that ${\| [x,y]\| = 0}$ for all ${x,y \in G}$, where ${[x,y] := xyx^{-1}y^{-1}}$ is the commutator.

We first establish some basic facts. Firstly, by hypothesis we have ${\|x^2\| = 2 \|x\|}$, and hence ${\|x^n \| = n \|x\|}$ whenever ${n}$ is a power of two. On the other hand, by the triangle inequality we have ${\|x^n \| \leq n\|x\|}$ for all positive ${n}$, and hence by the triangle inequality again we also have the matching lower bound, thus

$\displaystyle \|x^n \| = n \|x\|$

for all ${n > 0}$. The claim is also true for ${n=0}$ (apply the preceding bound with ${x=1}$ and ${n=2}$). By replacing ${\|x\|}$ with ${\max(\|x\|, \|x^{-1}\|)}$ if necessary we may now also assume without loss of generality that ${\|x^{-1} \| = \|x\|}$, thus

$\displaystyle \|x^n \| = |n| \|x\| \ \ \ \ \ (1)$

for all integers ${n}$.

Next, for any ${x,y \in G}$, and any natural number ${n}$, we have

$\displaystyle \|yxy^{-1} \| = \frac{1}{n} \| (yxy^{-1})^n \|$

$\displaystyle = \frac{1}{n} \| y x^n y^{-1} \|$

$\displaystyle \leq \frac{1}{n} ( \|y\| + n \|x\| + \|y\|^{-1} )$

so on taking limits as ${n \rightarrow \infty}$ we have ${\|yxy^{-1} \| \leq \|x\|}$. Replacing ${x,y}$ by ${yxy^{-1},y^{-1}}$ gives the matching lower bound, thus we have the conjugation invariance

$\displaystyle \|yxy^{-1} \| = \|x\|. \ \ \ \ \ (2)$

Next, we observe that if ${x,y,z,w}$ are such that ${x}$ is conjugate to both ${wy}$ and ${zw^{-1}}$, then one has the inequality

$\displaystyle \|x\| \leq \frac{1}{2} ( \|y \| + \| z \| ). \ \ \ \ \ (3)$

Indeed, if we write ${x = swys^{-1} = t zw^{-1} t^{-1}}$ for some ${s,t \in G}$, then for any natural number ${n}$ one has

$\displaystyle \|x\| = \frac{1}{2n} \| x^n x^n \|$

$\displaystyle = \frac{1}{2n} \| swy \dots wy s^{-1}t zw^{-1} \dots zw^{-1} t^{-1} \|$

where the ${wy}$ and ${zw^{-1}}$ terms each appear ${n}$ times. From (2) we see that conjugation by ${w}$ does not affect the norm. Using this and the triangle inequality several times, we conclude that

$\displaystyle \|x\| \leq \frac{1}{2n} ( \|s\| + n \|y\| + \| s^{-1} t\| + n \|z\| + \|t^{-1} \| ),$

and the claim (3) follows by sending ${n \rightarrow \infty}$.

The following special case of (3) will be of particular interest. Let ${x,y \in G}$, and for any integers ${m,k}$, define the quantity

$\displaystyle f(m,k) := \| x^m [x,y]^k \|.$

Observe that ${x^m [x,y]^k}$ is conjugate to both ${x (x^{m-1} [x,y]^k)}$ and to ${(y^{-1} x^m [x,y]^{k-1} xy) x^{-1}}$, hence by (3) one has

$\displaystyle \| x^m [x,y]^k \| \leq \frac{1}{2} ( \| x^{m-1} [x,y]^k \| + \| y^{-1} x^{m} [x,y]^{k-1} xy \|)$

which by (2) leads to the recursive inequality

$\displaystyle f(m,k) \leq \frac{1}{2} (f(m-1,k) + f(m+1,k-1)).$

We can write this in probabilistic notation as

$\displaystyle f(m,k) \leq {\bf E} f( (m,k) + X )$

where ${X}$ is a random vector that takes the values ${(-1,0)}$ and ${(1,-1)}$ with probability ${1/2}$ each. Iterating this, we conclude in particular that for any large natural number ${n}$, one has

$\displaystyle f(0,n) \leq {\bf E} f( Z )$

where ${Z := (0,n) + X_1 + \dots + X_{2n}}$ and ${X_1,\dots,X_{2n}}$ are iid copies of ${X}$. We can write ${Z = (1,-1/2) (Y_1 + \dots + Y_{2n})}$ where $Y_1,\dots,Y_{2n} = \pm 1$ are iid signs.  By the triangle inequality, we thus have

$\displaystyle f( Z ) \leq |Y_1+\dots+Y_{2n}| (\|x\| + \frac{1}{2} \| [x,y] \|),$

noting that $Y_1+\dots+Y_{2n}$ is an even integer.  On the other hand, $Y_1+\dots+Y_{2n}$ has mean zero and variance $2n$, hence by Cauchy-Schwarz

$\displaystyle f(0,n) \leq \sqrt{2n}( \|x\| + \frac{1}{2} \| [x,y] \|).$

But by (1), the left-hand side is equal to ${n \| [x,y]\|}$. Dividing by ${n}$ and then sending ${n \rightarrow \infty}$, we obtain the claim. $\Box$

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 ${G = (G,+)}$, thus for instance ${\| nx \| = |n| \|x\|}$ for ${n \in {\bf Z}}$. We think of ${G}$ as a ${{\bf Z}}$-module. One can then extend the seminorm to the associated ${{\bf Q}}$-vector space ${G \otimes_{\bf Z} {\bf Q}}$ by the formula ${\|\frac{a}{b} x\| := \frac{a}{b} \|x\|}$, and then to the associated ${{\bf R}}$-vector space ${G \otimes_{\bf Z} {\bf R}}$ by continuity, at which point it becomes a genuine seminorm (provided we have ensured the symmetry condition ${\|x\| = \|x^{-1}\|}$). Conversely, any seminorm on ${G \otimes_{\bf Z} {\bf R}}$ induces a seminorm on ${G}$. (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 ${F_2}$ be the free group on two generators ${a,b}$, and let ${\| \|: F_2 \rightarrow {\bf R}^+}$ be a quantity obeying the triangle inequality

$\displaystyle \| xy\| \leq \|x \| + \|y\|$

and the linear growth property

$\displaystyle \| x^n \| = |n| \| x\|$

for all ${x,y \in F_2}$ and integers ${n \in {\bf Z}}$; this implies the conjugation invariance

$\displaystyle \| y^{-1} x y \| = \|x\|$

or equivalently

$\displaystyle \| xy \| = \| yx\|$

We consider inequalities of the form

$\displaystyle \| xyx^{-1}y^{-1} \| \leq \alpha \|x\| + \beta \| y\| \ \ \ \ \ (1)$

or

$\displaystyle \| xyx^{-2}y^{-1} \| \leq \gamma \|x\| + \delta \| y\| \ \ \ \ \ (2)$

for various real numbers ${\alpha,\beta,\gamma,\delta}$. For instance, since

$\displaystyle \| xyx^{-1}y^{-1} \| \leq \| xyx^{-1}\| + \|y^{-1} \| = \|y\| + \|y\|$

we have (1) for ${(\alpha,\beta) = (2,0)}$. We also have the following further relations:

Proposition 1

• (i) If (1) holds for ${(\alpha,\beta)}$, then it holds for ${(\beta,\alpha)}$.
• (ii) If (1) holds for ${(\alpha,\beta)}$, then (2) holds for ${(\alpha+1, \frac{\beta}{2})}$.
• (iii) If (2) holds for ${(\gamma,\delta)}$, then (1) holds for ${(\frac{2\gamma}{3}, \frac{2\delta}{3})}$.
• (iv) If (1) holds for ${(\alpha,\beta)}$ and (2) holds for ${(\gamma,\delta)}$, then (1) holds for ${(\frac{2\alpha+1+\gamma}{4}, \frac{\delta+\beta}{4})}$.

Proof: For (i) we simply observe that

$\displaystyle \| xyx^{-1} y^{-1} \| = \| (xyx^{-1} y^{-1})^{-1} \| = \| y^{-1} x^{-1} y x \| = \| y x y^{-1} x^{-1} \|.$

For (ii), we calculate

$\displaystyle \| xyx^{-2}y^{-1} \| = \frac{1}{2}\| (xyx^{-2}y^{-1})^2 \|$

$\displaystyle = \frac{1}{2} \| (xyx^{-2}y^{-1} x) (yx^{-2} y^{-1}) \|$

$\displaystyle \leq \frac{1}{2} (\| xyx^{-2}y^{-1} x\| + \|yx^{-2} y^{-1}\|)$

$\displaystyle \leq \frac{1}{2} ( \| x^2 y x^{-2} y^{-1} \| + 2 \|x\| )$

$\displaystyle \leq \frac{1}{2} ( 2 \alpha \|x\| + \beta \|y\| + 2 \|x\|)$

giving the claim.

For (iii), we calculate

$\displaystyle \| xyx^{-1}y^{-1}\| = \frac{1}{3} \| (xyx^{-1}y^{-1})^3 \|$

$\displaystyle = \frac{1}{3} \| (xyx) (x^{-2} y^{-1} xy) (xyx)^{-1} (x^2 y x^{-1} y^{-1}) \|$

$\displaystyle \leq \frac{1}{3} ( \| x^{-2} y^{-1} xy\| + \| x^2 y x^{-1} y^{-1}\| )$

$\displaystyle = \frac{1}{3} ( \| xy x^{-2} y^{-1} \| + \|x^{-1} y^{-1} x^2 y \| )$

$\displaystyle \leq \frac{1}{3} ( \gamma \|x\| + \delta \|y\| + \gamma \|x\| + \delta \|y\|)$

giving the claim.

For (iv), we calculate

$\displaystyle \| xyx^{-1}y^{-1}\| = \frac{1}{4} \| (xyx^{-1}y^{-1})^4 \|$

$\displaystyle = \frac{1}{4} \| (xy) (x^{-1} y^{-1} x) (y x^{-1} y^{-1}) (xyx^{-1}) (xy)^{-1} (x^2yx^{-1}y^{-1}) \|$

$\displaystyle \leq \frac{1}{4} ( \| (x^{-1} y^{-1} x) (y x^{-1} y^{-1}) (xyx^{-1}) \| + \|x^2yx^{-1}y^{-1}\| )$

$\displaystyle \leq \frac{1}{4} ( \|(y x^{-1} y^{-1}) (xy^{-1}x^{-1})(x^{-1} y x) \| + \gamma \|x\| + \delta \|y\|)$

$\displaystyle \leq \frac{1}{4} ( \|x\| + \|(xy^{-1}x^{-1})(x^{-1} y x) \| + \gamma \|x\| + \delta \|y\|)$

$\displaystyle = \frac{1}{4} ( \|x\| + \|x^{-2} y x^2 y^{-1} \|+ \gamma \|x\| + \delta \|y\|)$

$\displaystyle \leq \frac{1}{4} ( \|x\| + 2\alpha \|x\| + \beta \|y\| + \gamma \|x\| + \delta \|y\|)$

giving the claim. $\Box$

Here is a typical application of the above estimates. If (1) holds for ${(\alpha,\beta)}$, then by part (i) it holds for ${(\beta,\alpha)}$, then by (ii) (2) holds for ${(\beta+1,\frac{\alpha}{2})}$, then by (iv) (1) holds for ${(\frac{3\beta+2}{4}, \frac{3\alpha}{8})}$. The map ${(\alpha,\beta) \mapsto (\frac{3\beta+2}{4}, \frac{3\alpha}{8})}$ has fixed point ${(\alpha,\beta) = (\frac{16}{23}, \frac{6}{23})}$, thus

$\displaystyle \| xyx^{-1}y^{-1} \| \leq \frac{16}{23} \|x\| + \frac{6}{23} \|y\|.$

For instance, if ${\|a\|, \|b\| \leq 1}$, then ${\|aba^{-1}b^{-1} \| \leq 22/23 = 0.95652\dots}$.

Here is a curious question posed to me by Apoorva Khare that I do not know the answer to. Let ${F_2}$ be the free group on two generators ${a,b}$. Does there exist a metric ${d}$ on this group which is

• bi-invariant, thus ${d(xg,yg)=d(gx,gy) = d(x,y)}$ for all ${x,y,g \in F_2}$; and
• linear growth in the sense that ${d(x^n,1) = n d(x,1)}$ for all ${x \in F_2}$ and all natural numbers ${n}$?

By defining the “norm” of an element ${x \in F_2}$ to be ${\| x\| := d(x,1)}$, an equivalent formulation of the problem asks if there exists a non-negative norm function ${\| \|: F_2 \rightarrow {\bf R}^+}$ that obeys the conjugation invariance

$\displaystyle \| gxg^{-1} \| = \|x \| \ \ \ \ \ (1)$

for all ${x,g \in F_2}$, the triangle inequality

$\displaystyle \| xy \| \leq \| x\| + \| y\| \ \ \ \ \ (2)$

for all ${x,y \in F_2}$, and the linear growth

$\displaystyle \| x^n \| = |n| \|x\| \ \ \ \ \ (3)$

for all ${x \in F_2}$ and ${n \in {\bf Z}}$, and such that ${\|x\| > 0}$ for all non-identity ${x \in F_2}$. Indeed, if such a norm exists then one can just take ${d(x,y) := \| x y^{-1} \|}$ to give the desired metric.

One can normalise the norm of the generators to be at most ${1}$, thus

$\displaystyle \| a \|, \| b \| \leq 1.$

This can then be used to upper bound the norm of other words in ${F_2}$. For instance, from (1), (3) one has

$\displaystyle \| aba^{-1} \|, \| b^{-1} a b \|, \| a^{-1} b^{-1} a \|, \| bab^{-1}\| \leq 1.$

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

$\displaystyle \| aba^{-1} b^{-1} \| = \frac{1}{3} \| (aba^{-1} b^{-1})^3 \|$

$\displaystyle = \frac{1}{3} \| (aba^{-1}) (b^{-1} ab) (a^{-1} b^{-1} a) (b ab^{-1}) \|$

$\displaystyle \leq \frac{4}{3}.$

In a similar spirit one has

$\displaystyle \| aba^{-2} b^{-1} \| = \frac{1}{2} \| (aba^{-2} b^{-1})^2 \|$

$\displaystyle = \frac{1}{2} \| (aba^{-1}) (a^{-1} b^{-1} a) (ba^{-1} b^{-1}) (ba^{-1} b^{-1}) \|$

$\displaystyle \leq 2.$

What is not clear to me is if one can keep arguing like this to continually improve the upper bounds on the norm ${\| g\|}$ of a given non-trivial group element ${g}$ to the point where this norm must in fact vanish, which would demonstrate that no metric with the above properties on ${F_2}$ 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

$\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) u_x = -\partial_x p \ \ \ \ \ (1)$

$\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) u_y = \rho - \partial_y p \ \ \ \ \ (2)$

$\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) \rho = 0 \ \ \ \ \ (3)$

$\displaystyle \partial_x u_x + \partial_y u_y = 0 \ \ \ \ \ (4)$

where ${u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2}$ is the velocity field, ${p: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ is the pressure field, and ${\rho: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ 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 ${D_t := \partial_t + u_x \partial_x + u_y \partial_y}$, one can abbreviate these equations as

$\displaystyle D_t u_x = -\partial_x p$

$\displaystyle D_t u_y = \rho - \partial_y p$

$\displaystyle D_t \rho = 0$

$\displaystyle \partial_x u_x + \partial_y u_y = 0.$

One can eliminate the role of the pressure ${p}$ by working with the vorticity ${\omega := \partial_x u_y - \partial_y u_x}$. A standard calculation then leads us to the equivalent “vorticity-stream” formulation

$\displaystyle D_t \omega = \partial_x \rho$

$\displaystyle D_t \rho = 0$

$\displaystyle \omega = \partial_x u_y - \partial_y u_x$

$\displaystyle \partial_x u_y + \partial_y u_y = 0$

of the Boussinesq equations. The latter two equations can be used to recover the velocity field ${u}$ from the vorticity ${\omega}$ by the Biot-Savart law

$\displaystyle u_x := -\partial_y \Delta^{-1} \omega; \quad u_y = \partial_x \Delta^{-1} \omega.$

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

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p$

$\displaystyle \nabla \cdot u = 0$

where now the velocity field ${u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3}$ and pressure field ${p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}}$ are over the three-dimensional domain ${{\bf R}^3}$. If one expresses ${{\bf R}^3}$ in polar coordinates ${(z,r,\theta)}$ then one can write the velocity vector field ${u}$ in these coordinates as

$\displaystyle u = u^z \frac{d}{dz} + u^r \frac{d}{dr} + u^\theta \frac{d}{d\theta}.$

If we make the axial symmetry assumption that these components, as well as ${p}$, do not depend on the ${\theta}$ variable, thus

$\displaystyle \partial_\theta u^z, \partial_\theta u^r, \partial_\theta u^\theta, \partial_\theta p = 0,$

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

$\displaystyle \tilde D_t \omega = \frac{1}{r^4} \partial_z \rho \ \ \ \ \ (5)$

$\displaystyle \tilde D_t \rho = 0 \ \ \ \ \ (6)$

$\displaystyle \omega = \frac{1}{r} (\partial_z u^r - \partial_r u^z) \ \ \ \ \ (7)$

$\displaystyle \partial_z(ru^z) + \partial_r(ru^r) = 0 \ \ \ \ \ (8)$

where ${\tilde D_t := \partial_t + u^z \partial_z + u^r \partial_r}$ is the modified material derivative, and ${\rho}$ is the field ${\rho := (r u^\theta)^2}$. This is almost identical with the Boussinesq equations except for some additional powers of ${r}$; thus, the intuition is that the Boussinesq equations are a simplified model for axially symmetric Euler flows when one stays away from the axis ${r=0}$ and also does not wander off to ${r=\infty}$.

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 ${r=1}$, 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

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

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

of the Euler equations to an arbitrary Riemannian manifold ${(M,g)}$ (ignoring any issues of boundary conditions for this discussion), where ${u: {\bf R} \rightarrow \Gamma(TM)}$ is a time-dependent vector field, ${p: {\bf R} \rightarrow C^\infty(M)}$ is a time-dependent scalar field, and ${\nabla_u}$ is the covariant derivative along ${u}$ using the Levi-Civita connection ${\nabla}$. In Penrose abstract index notation (using the Levi-Civita connection ${\nabla}$, and raising and lowering indices using the metric ${g = g_{ij}}$), the equations of motion become

$\displaystyle \partial_t u^i + u^j \nabla_j u^i = - \nabla^i p \ \ \ \ \ (9)$

$\displaystyle \nabla_i u^i = 0;$

in coordinates, this becomes

$\displaystyle \partial_t u^i + u^j (\partial_j u^i + \Gamma^i_{jk} u^k) = - g^{ij} \partial_j p$

$\displaystyle \partial_i u^i + \Gamma^i_{ik} u^k = 0 \ \ \ \ \ (10)$

where the Christoffel symbols ${\Gamma^i_{jk}}$ are given by the formula

$\displaystyle \Gamma^i_{jk} := \frac{1}{2} g^{il} (\partial_j g_{lk} + \partial_k g_{lj} - \partial_l g_{jk}),$

where ${g^{il}}$ is the inverse to the metric tensor ${g_{il}}$. If the coordinates are chosen so that the volume form ${dg}$ is the Euclidean volume form ${dx}$, thus ${\mathrm{det}(g)=1}$, then on differentiating we have ${g^{ij} \partial_k g_{ij} = 0}$, and hence ${\Gamma^i_{ik} = 0}$, and so the divergence-free equation (10) simplifies in this case to ${\partial_i u^i = 0}$. The Ebin-Marsden Euler equations are the natural generalisation of the Euler equations to arbitrary manifolds; for instance, they (formally) conserve the kinetic energy

$\displaystyle \frac{1}{2} \int_M |u|_g^2\ dg = \frac{1}{2} \int_M g_{ij} u^i u^j\ dg$

and 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 specific four-dimensional manifold in question is the space ${{\bf R} \times {\bf R}^+ \times {\bf R}/{\bf Z} \times {\bf R}/{\bf Z}}$ with metric

$\displaystyle dx^2 + dy^2 + y^{-1} dz^2 + y dw^2$

and solutions to the Boussinesq equation on ${{\bf R} \times {\bf R}^+}$ 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 ${\rho}$ appearing in the Boussinesq equations) can be incorporated into the dynamics via fluctuations in the Riemannian metric ${g}$). 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

$\displaystyle \mu(A+B) \geq \min(\mu(A)+\mu(B), 1) \ \ \ \ \ (1)$

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

$\displaystyle \mu(A), \mu(B), 1- \mu(A)-\mu(B) > 0. \ \ \ \ \ (2)$

The connectedness of ${G}$ is essential, otherwise one could form counterexamples involving proper subgroups of ${G}$ 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

$\displaystyle \mu( (A_1 \cup A_2) + B) + \mu( (A_1 \cap A_2) + B)$

$\displaystyle \leq \mu(A_1+B) + \mu(A_2+B) \ \ \ \ \ (3)$

which in turn easily follows from the identity ${(A_1 \cup A_2) + B = (A_1+B) \cup (A_2+B)}$ and the inclusion ${(A_1 \cap A_2) + B \subset (A_1 +B) \cap (A_2+B)}$, combined with the inclusion-exclusion formula.

In the non-trivial regime (2), equality can be attained in (1), for instance by taking ${G}$ to be the unit circle ${G = {\bf R}/{\bf Z}}$ and ${A,B}$ to be arcs in that circle (obeying (2)). A bit more generally, if ${G}$ is an arbitrary connected compact abelian group and ${\xi: G \rightarrow {\bf R}/{\bf Z}}$ is a non-trivial character (i.e., a continuous homomorphism), then ${\xi}$ must be surjective (as ${{\bf R}/{\bf Z}}$ has no non-trivial connected subgroups), and one can take ${A = \xi^{-1}(I)}$ and ${B = \xi^{-1}(J)}$ for some arcs ${I,J}$ 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 ${A,B}$ must be close (in measure) to an example of the above form ${A = \xi^{-1}(I), B = \xi^{-1}(J)}$. 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 ${A+B}$ is replaced by the partial sumset ${A +_\varepsilon B}$ consisting of “popular” sums).

Roughly speaking, the idea is as follows. Let us informally call ${(A,B)}$ 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 ${A,B}$, and invariant with respect to translation of either ${A}$ or ${B}$. Furthermore, from the submodularity inequality (3), one can show that if ${(A_1,B)}$ and ${(A_2,B)}$ are critical pairs (with ${\mu(A_1 \cap A_2)}$ and ${1 - \mu(A_1 \cup A_2) - \mu(B)}$ positive), then ${(A_1 \cap A_2,B)}$ and ${(A_1 \cup A_2, B)}$ are also critical pairs. (Note that this is consistent with the claim that critical pairs only occur when ${A,B}$ come from arcs of a circle.) Similarly, from associativity ${(A+B)+C = A+(B+C)}$, one can show that if ${(A,B)}$ and ${(A+B,C)}$ are critical pairs, then so are ${(B,C)}$ and ${(A,B+C)}$.

One can combine these closure properties to obtain further ones. For instance, suppose ${A,B}$ is such that ${\mu(A+B) 0}$. Then (cheating a little bit), one can show that ${(A+B,C)}$ is also a critical pair, basically because ${A+B}$ is the union of the ${A+b}$, ${b \in B}$, the ${(A+b,C)}$ are all critical pairs, and the ${A+b}$ 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 ${A+B}$ by the union of finitely many of the ${A+b}$, 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 ${(A,B)}$ and end up with a small set ${C}$ such that ${(A,C)}$ and ${(kC,C)}$ are also critical pairs for all ${1 \leq k \leq 10^4}$ (say), where ${kC}$ is the ${k}$-fold sumset of ${C}$. (Intuitively, if ${A,B}$ are thought of as secretly coming from the pullback of arcs ${I,J}$ by some character ${\xi}$, then ${C}$ should be the pullback of a much shorter arc by the same character.) In particular, ${C}$ exhibits linear growth, in that ${\mu(kC) = k\mu(C)}$ for all ${1 \leq k \leq 10^4}$. One can now use standard technology from inverse sumset theory to show first that ${C}$ has a very large Fourier coefficient (and thus is biased with respect to some character ${\xi}$), and secondly that ${C}$ is in fact almost of the form ${C = \xi^{-1}(K)}$ for some arc ${K}$, from which it is not difficult to conclude similar statements for ${A}$ and ${B}$ 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 ${f: {\bf N} \rightarrow {\bf C}}$ from the natural numbers to the complex numbers. Some fundamental examples of such functions include

• The constant function ${1: n \mapsto 1}$;
• The Kronecker delta function ${\delta: n \mapsto 1_{n=1}}$;
• The natural logarithm function ${L: n \mapsto \log n}$;
• The divisor function ${d_2: n \mapsto \sum_{d|n} 1}$;
• The von Mangoldt function ${\Lambda}$, with ${\Lambda(n)}$ defined to equal ${\log p}$ when ${n}$ is a power ${p^j}$ of a prime ${p}$ for some ${j \geq 1}$, and defined to equal zero otherwise; and
• The Möbius function ${\mu}$, with ${\mu(n)}$ defined to equal ${(-1)^k}$ when ${n}$ is the product of ${k}$ distinct primes, and defined to equal zero otherwise.

Given an arithmetic function ${f}$, we are often interested in statistics such as the summatory function

$\displaystyle \sum_{n \leq x} f(n), \ \ \ \ \ (1)$

the logarithmically (or harmonically) weighted summatory function

$\displaystyle \sum_{n \leq x} \frac{f(n)}{n}, \ \ \ \ \ (2)$

or the Dirichlet series

$\displaystyle {\mathcal D}[f](s) := \sum_n \frac{f(n)}{n^s}.$

In the latter case, one typically has to first restrict ${s}$ 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 ${\sum_{n \leq x} f(n) f(n+h)}$, 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 ${f,g: {\bf N} \rightarrow {\bf C}}$, forms a new arithmetic function ${f*g: {\bf N} \rightarrow {\bf C}}$, defined by the formula

$\displaystyle f*g(n) := \sum_{d|n} f(d) g(\frac{n}{d}).$

Thus for instance ${1*1 = d_2}$, ${1 * \Lambda = L}$, ${1 * \mu = \delta}$, and ${\delta * f = f}$ for any arithmetic function ${f}$. Dirichlet convolution and Dirichlet series are related by the fundamental formula

$\displaystyle {\mathcal D}[f * g](s) = {\mathcal D}[f](s) {\mathcal D}[g](s), \ \ \ \ \ (3)$

at least when the real part of ${s}$ 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

$\displaystyle {\mathcal D}[Lf](s) = - \frac{d}{ds} {\mathcal D}[f](s), \ \ \ \ \ (4)$

at least when the real part of ${s}$ 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 ${\zeta = {\mathcal D}[1]}$, thus for instance

$\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s)$

$\displaystyle {\mathcal D}[L](s) = - \zeta'(s)$

$\displaystyle {\mathcal D}[\delta](s) = 1$

$\displaystyle {\mathcal D}[\mu](s) = \frac{1}{\zeta(s)}$

$\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)}.$

Much of the difficulty of multiplicative number theory can be traced back to the discrete nature of the natural numbers ${{\bf N}}$, 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 ${{\bf N}_\infty := [1,+\infty)}$, 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 ${{\bf N}}$ at the infinite place ${\infty}$, hence the terminology.) Accordingly, let us define a continuous arithmetic function to be a locally integrable function ${f: {\bf N}_\infty \rightarrow {\bf C}}$. The analogue of the summatory function (1) is then an integral

$\displaystyle \int_1^x f(t)\ dt,$

and similarly the analogue of (2) is

$\displaystyle \int_1^x \frac{f(t)}{t}\ dt.$

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

$\displaystyle {\mathcal D}_\infty[f](s) := \int_1^\infty \frac{f(t)}{t^s}\ dt,$

which will be well-defined at least if the real part of ${s}$ is large enough and if the continuous arithmetic function ${f: {\bf N}_\infty \rightarrow {\bf C}}$ 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 ${1: {\bf N} \rightarrow {\bf C}}$ would be the constant function ${1_\infty: {\bf N}_\infty \rightarrow {\bf C}}$, which maps any ${t \in [1,+\infty)}$ to ${1}$, and which we will denote by ${1_\infty}$ in order to keep it distinct from ${1}$. The two functions ${1_\infty}$ and ${1}$ have approximately similar statistics; for instance one has

$\displaystyle \sum_{n \leq x} 1 = \lfloor x \rfloor \approx x-1 = \int_1^x 1\ dt$

and

$\displaystyle \sum_{n \leq x} \frac{1}{n} = H_{\lfloor x \rfloor} \approx \log x = \int_1^x \frac{1}{t}\ dt$

where ${H_n}$ is the ${n^{th}}$ harmonic number, and we are deliberately vague as to what the symbol ${\approx}$ means. Continuing this analogy, we would expect

$\displaystyle {\mathcal D}[1](s) = \zeta(s) \approx \frac{1}{s-1} = {\mathcal D}_\infty[1_\infty](s)$

which reflects the fact that ${\zeta}$ has a simple pole at ${s=1}$ with residue ${1}$, and no other poles. Note that the identity ${{\mathcal D}_\infty[1_\infty](s) = \frac{1}{s-1}}$ is initially only valid in the region ${\mathrm{Re} s > 1}$, but clearly the right-hand side can be continued analytically to the entire complex plane except for the pole at ${1}$, and so one can define ${{\mathcal D}_\infty[1_\infty]}$ in this region also.

In a similar vein, the logarithm function ${L: {\bf N} \rightarrow {\bf C}}$ is approximately similar to the logarithm function ${L_\infty: {\bf N}_\infty \rightarrow {\bf C}}$, giving for instance the crude form

$\displaystyle \sum_{n \leq x} L(n) = \log \lfloor x \rfloor! \approx x \log x - x = \int_1^\infty L_\infty(t)\ dt$

of Stirling’s formula, or the Dirichlet series approximation

$\displaystyle {\mathcal D}[L](s) = -\zeta'(s) \approx \frac{1}{(s-1)^2} = {\mathcal D}_\infty[L_\infty](s).$

The continuous analogue of Dirichlet convolution is multiplicative convolution using the multiplicative Haar measure ${\frac{dt}{t}}$: given two continuous arithmetic functions ${f_\infty, g_\infty: {\bf N}_\infty \rightarrow {\bf C}}$, one can define their convolution ${f_\infty *_\infty g_\infty: {\bf N}_\infty \rightarrow {\bf C}}$ by the formula

$\displaystyle f_\infty *_\infty g_\infty(t) := \int_1^t f_\infty(s) g_\infty(\frac{t}{s}) \frac{ds}{s}.$

Thus for instance ${1_\infty * 1_\infty = L_\infty}$. A short computation using Fubini’s theorem shows the analogue

$\displaystyle D_\infty[f_\infty *_\infty g_\infty](s) = D_\infty[f_\infty](s) D_\infty[g_\infty](s)$

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

$\displaystyle D_\infty[L_\infty f_\infty](s) = -\frac{d}{ds} D_\infty[f_\infty](s) \ \ \ \ \ (5)$

again assuming that the real part of ${s}$ 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 ${\rho}$, one has

$\displaystyle \frac{1}{s-\rho} = D_\infty[ t \mapsto t^{\rho-1} ](s)$

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

$\displaystyle \frac{1}{(s-\rho)^k} = D_\infty[ t \mapsto \frac{1}{(k-1)!} t^{\rho-1} \log^{k-1} t ](s)$

for any natural number ${k}$. This can lead to the following heuristic: if a Dirichlet series ${D[f](s)}$ behaves like a linear combination of poles ${\frac{1}{(s-\rho)^k}}$, in that

$\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{(s-\rho)^{k_\rho}}$

for some set ${\rho}$ of poles and some coefficients ${c_\rho}$ and natural numbers ${k_\rho}$ (where we again are vague as to what ${\approx}$ means, and how to interpret the sum ${\sum_\rho}$ if the set of poles is infinite), then one should expect the arithmetic function ${f}$ to behave like the continuous arithmetic function

$\displaystyle t \mapsto \sum_\rho \frac{c_\rho}{(k_\rho-1)!} t^{\rho-1} \log^{k_\rho-1} t.$

In particular, if we only have simple poles,

$\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{s-\rho}$

then we expect to have ${f}$ behave like continuous arithmetic function

$\displaystyle t \mapsto \sum_\rho c_\rho t^{\rho-1}.$

Integrating this from ${1}$ to ${x}$, this heuristically suggests an approximation

$\displaystyle \sum_{n \leq x} f(n) \approx \sum_\rho c_\rho \frac{x^\rho-1}{\rho}$

for the summatory function, and similarly

$\displaystyle \sum_{n \leq x} \frac{f(n)}{n} \approx \sum_\rho c_\rho \frac{x^{\rho-1}-1}{\rho-1},$

with the convention that ${\frac{x^\rho-1}{\rho}}$ is ${\log x}$ when ${\rho=0}$, and similarly ${\frac{x^{\rho-1}-1}{\rho-1}}$ is ${\log x}$ when ${\rho=1}$. 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

$\displaystyle \zeta(s) \approx \frac{1}{s-1} + \gamma$

to the zeta function near ${s=1}$, we have

$\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s) \approx \frac{1}{(s-1)^2} + \frac{2 \gamma}{s-1}$

we would expect that

$\displaystyle d_2 \approx L_\infty + 2 \gamma$

and thus for instance

$\displaystyle \sum_{n \leq x} d_2(n) \approx x \log x - x + 2 \gamma x$

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

Or, noting that ${\zeta(s)}$ has a simple pole at ${s=1}$ and assuming simple zeroes elsewhere, the log derivative ${-\zeta'(s)/\zeta(s)}$ will have simple poles of residue ${+1}$ at ${s=1}$ and ${-1}$ at all the zeroes, leading to the heuristic

$\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)} \approx \frac{1}{s-1} - \sum_\rho \frac{1}{s-\rho}$

suggesting that ${\Lambda}$ should behave like the continuous arithmetic function

$\displaystyle t \mapsto 1 - \sum_\rho t^{\rho-1}$

leading for instance to the summatory approximation

$\displaystyle \sum_{n \leq x} \Lambda(n) \approx x - \sum_\rho \frac{x^\rho-1}{\rho}$

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 1 Go 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 ${p}$-adic analogues of arithmetic functions to which a similar set of heuristics can be applied, perhaps to study sums such as ${\sum_{n \leq x: n = a \hbox{ mod } p^j} f(n)}$. A key problem here is that there does not seem to be any good interpretation of the expression ${\frac{1}{t^s}}$ when ${s}$ is complex and ${t}$ is a ${p}$-adic number, so it is not clear that one can analyse a Dirichlet series ${p}$-adically. For similar reasons, we don’t have a canonical way to define ${\chi(t)}$ for a Dirichlet character ${\chi}$ (unless its conductor happens to be a power of ${p}$), so there doesn’t seem to be much to say in the ${q}$-aspect either.

Alice Guionnet, Assaf Naor, Gilles Pisier, Sorin Popa, Dimitri Shylakhtenko, and I are organising a three month program here at the Institute for Pure and Applied Mathematics (IPAM) on the topic of Quantitative Linear Algebra.  The purpose of this program is to bring together mathematicians and computer scientists (both junior and senior) working in various quantitative aspects of linear operators, particularly in large finite dimension.  Such aspects include, but are not restricted to discrepancy theory, spectral graph theory, random matrices, geometric group theory, ergodic theory, von Neumann algebras, as well as specific research directions such as the Kadison-Singer problem, the Connes embedding conjecture and the Grothendieck inequality.  There will be several workshops and tutorials during the program (for instance I will be giving a series of introductory lectures on random matrix theory).

While we already have several confirmed participants, we are still accepting applications for this program until Dec 4; details of the application process may be found at this page.

In 2010, the UCLA mathematics department launched a scholarship opportunity for entering freshman students with exceptional background and promise in mathematics. We are able to offer one scholarship each year.  The UCLA Math Undergraduate Merit Scholarship provides for full tuition, and a room and board allowance for 4 years, contingent on continued high academic performance. In addition, scholarship recipients follow an individualized accelerated program of study, as determined after consultation with UCLA faculty.   The program of study leads to a Masters degree in Mathematics in four years.

More information and an application form for the scholarship can be found on the web at:

To be considered for Fall 2018, candidates must apply for the scholarship and also for admission to UCLA on or before November 30, 2017.

Let ${\lambda: {\bf N} \rightarrow \{-1,1\}}$ be the Liouville function, thus ${\lambda(n)}$ is defined to equal ${+1}$ when ${n}$ is the product of an even number of primes, and ${-1}$ when ${n}$ is the product of an odd number of primes. The Chowla conjecture asserts that ${\lambda}$ has the statistics of a random sign pattern, in the sense that

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) = 0 \ \ \ \ \ (1)$

for all ${k \geq 1}$ and all distinct natural numbers ${h_1,\dots,h_k}$, where we use the averaging notation

$\displaystyle \mathbb{E}_{n \leq N} f(n) := \frac{1}{N} \sum_{n \leq N} f(n).$

For ${k=1}$, this conjecture is equivalent to the prime number theorem (as discussed in this previous blog post), but the conjecture remains open for any ${k \geq 2}$.

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

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N}^{\log} \lambda(n+h_1) \dots \lambda(n+h_k) = 0 \ \ \ \ \ (2)$

of the conjecture, where we use the logarithmic averaging notation

$\displaystyle \mathbb{E}_{n \leq N}^{\log} f(n) := \frac{\sum_{n \leq N} \frac{f(n)}{n}}{\sum_{n \leq N} \frac{1}{n}}.$

Using the summation by parts (or telescoping series) identity

$\displaystyle \sum_{n \leq N} \frac{f(n)}{n} = \sum_{M < N} \frac{1}{M(M+1)} (\sum_{n \leq M} f(n)) + \frac{1}{N} \sum_{n \leq N} f(n) \ \ \ \ \ (3)$

it is not difficult to show that the Chowla conjecture (1) for a given ${k,h_1,\dots,h_k}$ implies the logarithmically averaged conjecture (2). However, the converse implication is not at all clear. For instance, for ${k=1}$, we have already mentioned that the Chowla conjecture

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N} \lambda(n) = 0$

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

$\displaystyle \lim_{N \rightarrow \infty} \mathbb{E}^{\log}_{n \leq N} \lambda(n) = 0$

is significantly easier to show (a proof with the Liouville function ${\lambda}$ replaced by the closely related Möbius function ${\mu}$ 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 ${k=2}$, and in this recent paper with Joni Teravainen, we proved the conjecture for all odd ${k}$ (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 1 Assume that the logarithmically averaged Chowla conjecture (2) is true for all ${k}$. Then there exists a sequence ${N_i}$ going to infinity such that the Chowla conjecture (1) is true for all ${k}$ along that sequence, that is to say

$\displaystyle \lim_{N_i \rightarrow \infty} \mathbb{E}_{n \leq N_i} \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

for all ${k}$ and all distinct ${h_1,\dots,h_k}$.

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 2 Let ${k}$ be a natural number. Assume that the logarithmically averaged Chowla conjecture (2) is true for ${2k}$. Then there exists a set ${{\mathcal N}}$ of natural numbers of logarithmic density ${1}$ (that is, ${\lim_{N \rightarrow \infty} \mathbb{E}_{n \leq N}^{\log} 1_{n \in {\mathcal N}} = 1}$) such that

$\displaystyle \lim_{N \rightarrow \infty: N \in {\mathcal N}} \mathbb{E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

for any distinct ${h_1,\dots,h_k}$.

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 (${k=2}$ and odd ${k}$) 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 ${h_1,\dots,h_k}$, we take a large number ${H}$ and consider the limiting the second moment

$\displaystyle \limsup_{N \rightarrow \infty} \mathop{\bf E}_{n \leq N}^{\log} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2.$

We can expand this as

$\displaystyle \limsup_{N \rightarrow \infty} \mathop{\bf E}_{m,m' \leq H} \mathop{\bf E}_{n \leq N}^{\log} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)$

$\displaystyle \lambda(n+m'+h_1) \dots \lambda(n+m'+h_k).$

If all the ${m+h_1,\dots,m+h_k,m'+h_1,\dots,m'+h_k}$ are distinct, the hypothesis (2) tells us that the inner averages goes to zero as ${N \rightarrow \infty}$. The remaining averages are ${O(1)}$, and there are ${O( k^2 )}$ of these averages. We conclude that

$\displaystyle \limsup_{N \rightarrow \infty} \mathop{\bf E}_{n \leq N}^{\log} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2 \ll k^2 / H.$

By Markov’s inequality (and (3)), we conclude that for any fixed ${h_1,\dots,h_k, H}$, there exists a set ${{\mathcal N}_{h_1,\dots,h_k,H}}$ of upper logarithmic density at least ${1-k/H^{1/2}}$, thus

$\displaystyle \limsup_{N \rightarrow \infty} \mathbb{E}_{n \leq N}^{\log} 1_{n \in {\mathcal N}_{h_1,\dots,h_k,H}} \geq 1 - k/H^{1/2}$

such that

$\displaystyle \mathop{\bf E}_{n \leq N} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2 \ll k / H^{1/2}.$

By deleting at most finitely many elements, we may assume that ${{\mathcal N}_{h_1,\dots,h_k,H}}$ consists only of elements of size at least ${H^2}$ (say).

For any ${H_0}$, if we let ${{\mathcal N}_{h_1,\dots,h_k, \geq H_0}}$ be the union of ${{\mathcal N}_{h_1,\dots,h_k, H}}$ for ${H \geq H_0}$, then ${{\mathcal N}_{h_1,\dots,h_k, \geq H_0}}$ has logarithmic density ${1}$. By a diagonalisation argument (using the fact that the set of tuples ${(h_1,\dots,h_k)}$ is countable), we can then find a set ${{\mathcal N}}$ of natural numbers of logarithmic density ${1}$, such that for every ${h_1,\dots,h_k,H_0}$, every sufficiently large element of ${{\mathcal N}}$ lies in ${{\mathcal N}_{h_1,\dots,h_k,\geq H_0}}$. Thus for every sufficiently large ${N}$ in ${{\mathcal N}}$, one has

$\displaystyle \mathop{\bf E}_{n \leq N} |\mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k)|^2 \ll k / H^{1/2}.$

for some ${H \geq H_0}$ with ${N \geq H^2}$. By Cauchy-Schwarz, this implies that

$\displaystyle \mathop{\bf E}_{n \leq N} \mathop{\bf E}_{m \leq H} \lambda(n+m+h_1) \dots \lambda(n+m+h_k) \ll k^{1/2} / H^{1/4};$

interchanging the sums and using ${N \geq H^2}$ and ${H \geq H_0}$, this implies that

$\displaystyle \mathop{\bf E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) \ll k^{1/2} / H^{1/4} \leq k^{1/2} / H_0^{1/4}.$

We conclude on taking ${H_0}$ to infinity that

$\displaystyle \lim_{N \rightarrow \infty; N \in {\mathcal N}} \mathop{\bf E}_{n \leq N} \lambda(n+h_1) \dots \lambda(n+h_k) = 0$

as required.