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The Euler equations for three-dimensional incompressible inviscid fluid flow are

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

where ${u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3}$ is the velocity field, and ${p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}}$ is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as ${(-\Delta)^{-1}}$ or ${|\nabla|^{-1} := (-\Delta)^{-1/2}}$ formally, assuming that these inverses are well defined on the functions they are applied to.

Meanwhile, the surface quasi-geostrophic (SQG) equation is given by

$\displaystyle \partial_t \theta + (u \cdot \nabla) \theta = 0 \ \ \ \ \ (2)$

$\displaystyle u = ( -\partial_y |\nabla|^{-1}, \partial_x |\nabla|^{-1} ) \theta \ \ \ \ \ (3)$

where ${\theta: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ is the active scalar, and ${u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2}$ is the velocity field. The SQG equations are often used as a toy model for the 3D Euler equations, as they share many of the same features (e.g. vortex stretching); see this paper of Constantin, Majda, and Tabak for more discussion (or this previous blog post).

I recently found a more direct way to connect the two equations. We first recall that the Euler equations can be placed in vorticity-stream form by focusing on the vorticity ${\omega := \nabla \times u}$. Indeed, taking the curl of (1), we obtain the vorticity equation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (4)$

while the velocity ${u}$ can be recovered from the vorticity via the Biot-Savart law

$\displaystyle u = (-\Delta)^{-1} \nabla \times \omega. \ \ \ \ \ (5)$

The system (4), (5) has some features in common with the system (2), (3); in (2) it is a scalar field ${\theta}$ that is being transported by a divergence-free vector field ${u}$, which is a linear function of the scalar field as per (3), whereas in (4) it is a vector field ${\omega}$ that is being transported (in the Lie derivative sense) by a divergence-free vector field ${u}$, which is a linear function of the vector field as per (5). However, the system (4), (5) is in three dimensions whilst (2), (3) is in two spatial dimensions, the dynamical field is a scalar field ${\theta}$ for SQG and a vector field ${\omega}$ for Euler, and the relationship between the velocity field and the dynamical field is given by a zeroth order Fourier multiplier in (3) and a ${-1^{th}}$ order operator in (5).

However, we can make the two equations more closely resemble each other as follows. We first consider the generalisation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (6)$

$\displaystyle u = T (-\Delta)^{-1} \nabla \times \omega \ \ \ \ \ (7)$

where ${T}$ is an invertible, self-adjoint, positive-definite zeroth order Fourier multiplier that maps divergence-free vector fields to divergence-free vector fields. The Euler equations then correspond to the case when ${T}$ is the identity operator. As discussed in this previous blog post (which used ${A}$ to denote the inverse of the operator denoted here as ${T}$), this generalised Euler system has many of the same features as the original Euler equation, such as a conserved Hamiltonian

$\displaystyle \frac{1}{2} \int_{{\bf R}^3} u \cdot T^{-1} u,$

the Kelvin circulation theorem, and conservation of helicity

$\displaystyle \int_{{\bf R}^3} \omega \cdot T^{-1} u.$

Also, if we require ${\omega}$ to be divergence-free at time zero, it remains divergence-free at all later times.

Let us consider “two-and-a-half-dimensional” solutions to the system (6), (7), in which ${u,\omega}$ do not depend on the vertical coordinate ${z}$, thus

$\displaystyle \omega(t,x,y,z) = \omega(t,x,y)$

and

$\displaystyle u(t,x,y,z) = u(t,x,y)$

but we allow the vertical components ${u_z, \omega_z}$ to be non-zero. For this to be consistent, we also require ${T}$ to commute with translations in the ${z}$ direction. As all derivatives in the ${z}$ direction now vanish, we can simplify (6) to

$\displaystyle D_t \omega = (\omega_x \partial_x + \omega_y \partial_y) u \ \ \ \ \ (8)$

where ${D_t}$ is the two-dimensional material derivative

$\displaystyle D_t := \partial_t + u_x \partial_x + u_y \partial_y.$

Also, divergence-free nature of ${\omega,u}$ then becomes

$\displaystyle \partial_x \omega_x + \partial_y \omega_y = 0$

and

$\displaystyle \partial_x u_x + \partial_y u_y = 0. \ \ \ \ \ (9)$

In particular, we may (formally, at least) write

$\displaystyle (\omega_x, \omega_y) = (\partial_y \theta, -\partial_x \theta)$

for some scalar field ${\theta(t,x,y,z) = \theta(t,x,y)}$, so that (7) becomes

$\displaystyle u = T ( (- \Delta)^{-1} \partial_y \omega_z, - (-\Delta^{-1}) \partial_x \omega_z, \theta ). \ \ \ \ \ (10)$

The first two components of (8) become

$\displaystyle D_t \partial_y \theta = \partial_y \theta \partial_x u_x - \partial_x \theta \partial_y u_x$

$\displaystyle - D_t \partial_x \theta = \partial_y \theta \partial_x u_y - \partial_x \theta \partial_y u_y$

which rearranges using (9) to

$\displaystyle \partial_y D_t \theta = \partial_x D_t \theta = 0.$

Formally, we may integrate this system to obtain the transport equation

$\displaystyle D_t \theta = 0, \ \ \ \ \ (11)$

Finally, the last component of (8) is

$\displaystyle D_t \omega_z = \partial_y \theta \partial_x u_z - \partial_x \theta \partial_y u_z. \ \ \ \ \ (12)$

At this point, we make the following choice for ${T}$:

$\displaystyle T ( U_x, U_y, \theta ) = \alpha (U_x, U_y, \theta) + (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, 0) \ \ \ \ \ (13)$

$\displaystyle + P( 0, 0, |\nabla|^{-1} (\partial_y U_x - \partial_x U_y) )$

where ${\alpha > 0}$ is a real constant and ${Pu := (-\Delta)^{-1} (\nabla \times (\nabla \times u))}$ is the Leray projection onto divergence-free vector fields. One can verify that for large enough ${\alpha}$, ${T}$ is a self-adjoint positive definite zeroth order Fourier multiplier from divergence free vector fields to divergence-free vector fields. With this choice, we see from (10) that

$\displaystyle u_z = \alpha \theta - |\nabla|^{-1} \omega_z$

so that (12) simplifies to

$\displaystyle D_t \omega_z = - \partial_y \theta \partial_x |\nabla|^{-1} \omega_z + \partial_x \theta \partial_y |\nabla|^{-1} \omega_z.$

This implies (formally at least) that if ${\omega_z}$ vanishes at time zero, then it vanishes for all time. Setting ${\omega_z=0}$, we then have from (10) that

$\displaystyle (u_x,u_y,u_z) = (-\partial_y |\nabla|^{-1} \theta, \partial_x |\nabla|^{-1} \theta, \alpha \theta )$

and from (11) we then recover the SQG system (2), (3). To put it another way, if ${\theta(t,x,y)}$ and ${u(t,x,y)}$ solve the SQG system, then by setting

$\displaystyle \omega(t,x,y,z) := ( \partial_y \theta(t,x,y), -\partial_x \theta(t,x,y), 0 )$

$\displaystyle \tilde u(t,x,y,z) := ( u_x(t,x,y), u_y(t,x,y), \alpha \theta(t,x,y) )$

then ${\omega,\tilde u}$ solve the modified Euler system (6), (7) with ${T}$ given by (13).

We have ${T^{-1} \tilde u = (0, 0, \theta)}$, so the Hamiltonian ${\frac{1}{2} \int_{{\bf R}^3} \tilde u \cdot T^{-1} \tilde u}$ for the modified Euler system in this case is formally a scalar multiple of the conserved quantity ${\int_{{\bf R}^2} \theta^2}$. The momentum ${\int_{{\bf R}^3} x \cdot \tilde u}$ for the modified Euler system is formally a scalar multiple of the conserved quantity ${\int_{{\bf R}^2} \theta}$, while the vortex stream lines that are preserved by the modified Euler flow become the level sets of the active scalar that are preserved by the SQG flow. On the other hand, the helicity ${\int_{{\bf R}^3} \omega \cdot T^{-1} \tilde u}$ vanishes, and other conserved quantities for SQG (such as the Hamiltonian ${\int_{{\bf R}^2} \theta |\nabla|^{-1} \theta}$) do not seem to correspond to conserved quantities of the modified Euler system. This is not terribly surprising; a low-dimensional flow may well have a richer family of conservation laws than the higher-dimensional system that it is embedded in.

An extremely large portion of mathematics is concerned with locating solutions to equations such as

$\displaystyle f(x) = 0$

or

$\displaystyle \Phi(x) = x \ \ \ \ \ (1)$

for ${x}$ in some suitable domain space (either finite-dimensional or infinite-dimensional), and various maps ${f}$ or ${\Phi}$. To solve the fixed point iteration equation (1), the simplest general method available is the fixed point iteration method: one starts with an initial approximate solution ${x_0}$ to (1), so that ${\Phi(x_0) \approx x_0}$, and then recursively constructs the sequence ${x_1, x_2, x_3, \dots}$ by ${x_n := \Phi(x_{n-1})}$. If ${\Phi}$ behaves enough like a “contraction”, and the domain is complete, then one can expect the ${x_n}$ to converge to a limit ${x}$, which should then be a solution to (1). For instance, if ${\Phi: X \rightarrow X}$ is a map from a metric space ${X = (X,d)}$ to itself, which is a contraction in the sense that

$\displaystyle d( \Phi(x), \Phi(y) ) \leq (1-\eta) d(x,y)$

for all ${x,y \in X}$ and some ${\eta>0}$, then with ${x_n}$ as above we have

$\displaystyle d( x_{n+1}, x_n ) \leq (1-\eta) d(x_n, x_{n-1} )$

for any ${n}$, and so the distances ${d(x_n, x_{n-1} )}$ between successive elements of the sequence decay at at least a geometric rate. This leads to the contraction mapping theorem, which has many important consequences, such as the inverse function theorem and the Picard existence theorem.

A slightly more complicated instance of this strategy arises when trying to linearise a complex map ${f: U \rightarrow {\bf C}}$ defined in a neighbourhood ${U}$ of a fixed point. For simplicity we normalise the fixed point to be the origin, thus ${0 \in U}$ and ${f(0)=0}$. When studying the complex dynamics ${f^2 = f \circ f}$, ${f^3 = f \circ f \circ f}$, ${\dots}$ of such a map, it can be useful to try to conjugate ${f}$ to another function ${g = \psi^{-1} \circ f \circ \psi}$, where ${\psi}$ is a holomorphic function defined and invertible near ${0}$ with ${\psi(0)=0}$, since the dynamics of ${g}$ will be conjguate to that of ${f}$. Note that if ${f(0)=0}$ and ${f'(0)=\lambda}$, then from the chain rule any conjugate ${g}$ of ${f}$ will also have ${g(0)=0}$ and ${g'(0)=\lambda}$. Thus, the “simplest” function one can hope to conjugate ${f}$ to is the linear function ${z \mapsto \lambda z}$. Let us say that ${f}$ is linearisable (around ${0}$) if it is conjugate to ${z \mapsto \lambda z}$ in some neighbourhood of ${0}$. Equivalently, ${f}$ is linearisable if there is a solution to the Schröder equation

$\displaystyle f( \psi(z) ) = \psi(\lambda z) \ \ \ \ \ (2)$

for some ${\psi: U' \rightarrow {\bf C}}$ defined and invertible in a neighbourhood ${U'}$ of ${0}$ with ${\psi(0)=0}$, and all ${z}$ sufficiently close to ${0}$. (The Schröder equation is normalised somewhat differently in the literature, but this form is equivalent to the usual form, at least when ${\lambda}$ is non-zero.) Note that if ${\psi}$ solves the above equation, then so does ${z \mapsto \psi(cz)}$ for any non-zero ${c}$, so we may normalise ${\psi'(0)=1}$ in addition to ${\psi(0)=0}$, which also ensures local invertibility from the inverse function theorem. (Note from winding number considerations that ${\psi}$ cannot be invertible near zero if ${\psi'(0)}$ vanishes.)

We have the following basic result of Koenigs:

Theorem 1 (Koenig’s linearisation theorem) Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function defined near ${0}$ with ${f(0)=0}$ and ${f'(0)=\lambda}$. If ${0 < |\lambda| < 1}$ (attracting case) or ${1 < |\lambda| < \infty}$ (repelling case), then ${f}$ is linearisable near zero.

Proof: Observe that if ${f, \psi, \lambda}$ solve (2), then ${f^{-1}, \psi^{-1}, \lambda^{-1}}$ solve (2) also (in a sufficiently small neighbourhood of zero). Thus we may reduce to the attractive case ${0 < |\lambda| < 1}$.

Let ${r>0}$ be a sufficiently small radius, and let ${X}$ denote the space of holomorphic functions ${\psi: B(0,r) \rightarrow {\bf C}}$ on the complex disk ${B(0,r) := \{z \in {\bf C}: |z| < r \}}$ with ${\psi(0)=0}$ and ${\psi'(0)=1}$. We can view the Schröder equation (2) as a fixed point equation

$\displaystyle \psi = \Phi(\psi)$

where ${\Phi: X' \rightarrow X}$ is the partially defined function on ${X}$ that maps a function ${\psi: B(0,r) \rightarrow {\bf C}}$ to the function ${\Phi(\psi): B(0,r) \rightarrow {\bf C}}$ defined by

$\displaystyle \Phi(\psi)(z) := f^{-1}( \psi( \lambda z ) ),$

assuming that ${f^{-1}}$ is well-defined on the range of ${\psi(B(0,\lambda r))}$ (this is why ${\Phi}$ is only partially defined).

We can solve this equation by the fixed point iteration method, if ${r}$ is small enough. Namely, we start with ${\psi_0: B(0,r) \rightarrow {\bf C}}$ being the identity map, and set ${\psi_1 := \Phi(\psi_0), \psi_2 := \Phi(\psi_1)}$, etc. We equip ${X}$ with the uniform metric ${d( \psi, \tilde \psi ) := \sup_{z \in B(0,r)} |\psi(z) - \tilde \psi(z)|}$. Observe that if ${d( \psi, \psi_0 ), d(\tilde \psi, \psi_0) \leq r}$, and ${r}$ is small enough, then ${\psi, \tilde \psi}$ takes values in ${B(0,2r)}$, and ${\Phi(\psi), \Phi(\tilde \psi)}$ are well-defined and lie in ${X}$. Also, since ${f^{-1}}$ is smooth and has derivative ${\lambda^{-1}}$ at ${0}$, we have

$\displaystyle |f^{-1}(z) - f^{-1}(w)| \leq (1+\varepsilon) |\lambda|^{-1} |z-w|$

if ${z, w \in B(0,r)}$, ${\varepsilon>0}$ and ${r}$ is sufficiently small depending on ${\varepsilon}$. This is not yet enough to establish the required contraction (thanks to Mario Bonk for pointing this out); but observe that the function ${\frac{\psi(z)-\tilde \psi(z)}{z^2}}$ is holomorphic on ${B(0,r)}$ and bounded by ${d(\psi,\tilde \psi)/r^2}$ on the boundary of this ball (or slightly within this boundary), so by the maximum principle we see that

$\displaystyle |\frac{\psi(z)-\tilde \psi(z)}{z^2}| \leq \frac{1}{r^2} d(\psi,\tilde \psi)$

on all of ${B(0,r)}$, and in particular

$\displaystyle |\psi(z)-\tilde \psi(z)| \leq |\lambda|^2 d(\psi,\tilde \psi)$

on ${B(0,\lambda r)}$. Putting all this together, we see that

$\displaystyle d( \Phi(\psi), \Phi(\tilde \psi)) \leq (1+\varepsilon) |\lambda| d(\psi, \tilde \psi);$

since ${|\lambda|<1}$, we thus obtain a contraction on the ball ${\{ \psi \in X: d(\psi,\psi_0) \leq r \}}$ if ${\varepsilon}$ is small enough (and ${r}$ sufficiently small depending on ${\varepsilon}$). From this (and the completeness of ${X}$, which follows from Morera’s theorem) we see that the iteration ${\psi_n}$ converges (exponentially fast) to a limit ${\psi \in X}$ which is a fixed point of ${\Phi}$, and thus solves Schröder’s equation, as required. $\Box$

Koenig’s linearisation theorem leaves open the indifferent case when ${|\lambda|=1}$. In the rationally indifferent case when ${\lambda^n=1}$ for some natural number ${n}$, there is an obvious obstruction to linearisability, namely that ${f^n = 1}$ (in particular, linearisation is not possible in this case when ${f}$ is a non-trivial rational function). An obstruction is also present in some irrationally indifferent cases (where ${|\lambda|=1}$ but ${\lambda^n \neq 1}$ for any natural number ${n}$), if ${\lambda}$ is sufficiently close to various roots of unity; the first result of this form is due to Cremer, and the optimal result of this type for quadratic maps was established by Yoccoz. In the other direction, we have the following result of Siegel:

Theorem 2 (Siegel’s linearisation theorem) Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function defined near ${0}$ with ${f(0)=0}$ and ${f'(0)=\lambda}$. If ${|\lambda|=1}$ and one has the Diophantine condition ${\frac{1}{|\lambda^n-1|} \leq C n^C}$ for all natural numbers ${n}$ and some constant ${C>0}$, then ${f}$ is linearisable at ${0}$.

The Diophantine condition can be relaxed to a more general condition involving the rational exponents of the phase ${\theta}$ of ${\lambda = e^{2\pi i \theta}}$; this was worked out by Brjuno, with the condition matching the one later obtained by Yoccoz. Amusingly, while the set of Diophantine numbers (and hence the set of linearisable ${\lambda}$) has full measure on the unit circle, the set of non-linearisable ${\lambda}$ is generic (the complement of countably many nowhere dense sets) due to the above-mentioned work of Cremer, leading to a striking disparity between the measure-theoretic and category notions of “largeness”.

Siegel’s theorem does not seem to be provable using a fixed point iteration method. However, it can be established by modifying another basic method to solve equations, namely Newton’s method. Let us first review how this method works to solve the equation ${f(x)=0}$ for some smooth function ${f: I \rightarrow {\bf R}}$ defined on an interval ${I}$. We suppose we have some initial approximant ${x_0 \in I}$ to this equation, with ${f(x_0)}$ small but not necessarily zero. To make the analysis more quantitative, let us suppose that the interval ${[x_0-r_0,x_0+r_0]}$ lies in ${I}$ for some ${r_0>0}$, and we have the estimates

$\displaystyle |f(x_0)| \leq \delta_0 r_0$

$\displaystyle |f'(x)| \geq \eta_0$

$\displaystyle |f''(x)| \leq \frac{1}{\eta_0 r_0}$

for some ${\delta_0 > 0}$ and ${0 < \eta_0 < 1/2}$ and all ${x \in [x_0-r_0,x_0+r_0]}$ (the factors of ${r_0}$ are present to make ${\delta_0,\eta_0}$ “dimensionless”).

Lemma 3 Under the above hypotheses, we can find ${x_1}$ with ${|x_1 - x_0| \leq \eta_0 r_0}$ such that

$\displaystyle |f(x_1)| \ll \delta_0^2 \eta_0^{-O(1)} r_0.$

In particular, setting ${r_1 := (1-\eta_0) r_0}$, ${\eta_1 := \eta_0/2}$, and ${\delta_1 = O(\delta_0^2 \eta_0^{-O(1)})}$, we have ${[x_1-r_1,x_1+r_1] \subset [x_0-r_0,x_0+r_0] \subset I}$, and

$\displaystyle |f(x_1)| \leq \delta_1 r_1$

$\displaystyle |f'(x)| \geq \eta_1$

$\displaystyle |f''(x)| \leq \frac{1}{\eta_1 r_1}$

for all ${x \in [x_1-r_1,x_1+r_1]}$.

The crucial point here is that the new error ${\delta_1}$ is roughly the square of the previous error ${\delta_0}$. This leads to extremely fast (double-exponential) improvement in the error upon iteration, which is more than enough to absorb the exponential losses coming from the ${\eta_0^{-O(1)}}$ factor.

Proof: If ${\delta_0 > c \eta_0^{C}}$ for some absolute constants ${C,c>0}$ then we may simply take ${x_0=x_1}$, so we may assume that ${\delta_0 \leq c \eta_0^{C}}$ for some small ${c>0}$ and large ${C>0}$. Using the Newton approximation ${f(x_0+h) \approx f(x_0) + h f'(x_0)}$ we are led to the choice

$\displaystyle x_1 := x_0 - \frac{f(x_0)}{f'(x_0)}$

for ${x_1}$. From the hypotheses on ${f}$ and the smallness hypothesis on ${\delta}$ we certainly have ${|x_1-x_0| \leq \eta_0 r_0}$. From Taylor’s theorem with remainder we have

$\displaystyle f(x_1) = f(x_0) - \frac{f(x_0)}{f'(x_0)} f'(x_0) + O( \frac{1}{\eta_0 r_0} |\frac{f(x_0)}{f'(x_0)}|^2 )$

$\displaystyle = O( \frac{1}{\eta_0 r_0} (\frac{\delta_0 r_0}{\eta_0})^2 )$

and the claim follows. $\Box$

We can iterate this procedure; starting with ${x_0,\eta_0,r_0,\delta_0}$ as above, we obtain a sequence of nested intervals ${[x_n-r_n,x_n+r_n]}$ with ${f(x_n)| \leq \delta_n}$, and with ${\eta_n,r_n,\delta_n,x_n}$ evolving by the recursive equations and estimates

$\displaystyle \eta_n = \eta_{n-1} / 2$

$\displaystyle r_n = (1 - \eta_{n-1}) r_{n-1}$

$\displaystyle \delta_n = O( \delta_{n-1}^2 \eta_{n-1}^{-O(1)} )$

$\displaystyle |x_n - x_{n-1}| \leq \eta_{n-1} r_{n-1}.$

If ${\delta_0}$ is sufficiently small depending on ${\eta_0}$, we see that ${\delta_n}$ converges rapidly to zero (indeed, we can inductively obtain a bound of the form ${\delta_n \leq \eta_0^{C (2^n + n)}}$ for some large absolute constant ${C}$ if ${\delta_0}$ is small enough), and ${x_n}$ converges to a limit ${x \in I}$ which then solves the equation ${f(x)=0}$ by the continuity of ${f}$.

As I recently learned from Zhiqiang Li, a similar scheme works to prove Siegel’s theorem, as can be found for instance in this text of Carleson and Gamelin. The key is the following analogue of Lemma 3.

Lemma 4 Let ${\lambda}$ be a complex number with ${|\lambda|=1}$ and ${\frac{1}{|\lambda^n-1|} \ll n^{O(1)}}$ for all natural numbers ${n}$. Let ${r_0>0}$, and let ${f_0: B(0,r_0) \rightarrow {\bf C}}$ be a holomorphic function with ${f_0(0)=0}$, ${f'_0(0)=\lambda}$, and

$\displaystyle |f_0(z) - \lambda z| \leq \delta_0 r_0 \ \ \ \ \ (3)$

for all ${z \in B(0,r_0)}$ and some ${\delta_0>0}$. Let ${0 < \eta_0 \leq 1/2}$, and set ${r_1 := (1-\eta_0) r_0}$. Then there exists an injective holomorphic function ${\psi_0: B(0, r_1) \rightarrow B(0, r_0)}$ and a holomorphic function ${f_1: B(0,r_1) \rightarrow {\bf C}}$ such that

$\displaystyle f_0( \psi_1(z) ) = \psi_1(f_1(z)) \ \ \ \ \ (4)$

for all ${z \in B(0,r_1)}$, and such that

$\displaystyle |\psi_1(z) - z| \ll \delta_0 \eta_0^{-O(1)} r_1$

and

$\displaystyle |f_1(z) - \lambda z| \leq \delta_1 r_1$

for all ${z \in B(0,r_1)}$ and some ${\delta_1 = O(\delta_0^2 \eta_0^{-O(1)})}$.

Proof: By scaling we may normalise ${r_0=1}$. If ${\delta_0 > c \eta_0^C}$ for some constants ${c,C>0}$, then we can simply take ${\psi_1}$ to be the identity and ${f_1=f_0}$, so we may assume that ${\delta_0 \leq c \eta_0^C}$ for some small ${c>0}$ and large ${C>0}$.

To motivate the choice of ${\psi_1}$, we write ${f_0(z) = \lambda z + \hat f_0(z)}$ and ${\psi_1(z) = z + \hat \psi(z)}$, with ${\hat f_0}$ and ${\hat \psi_1}$ viewed as small. We would like to have ${f_0(\psi_1(z)) \approx \psi_1(\lambda z)}$, which expands as

$\displaystyle \lambda z + \lambda \hat \psi_1(z) + \hat f_0( z + \hat \psi_1(z) ) \approx \lambda z + \hat \psi_1(\lambda z).$

As ${\hat f_0}$ and ${\hat \psi}$ are both small, we can heuristically approximate ${\hat f_0(z + \hat \psi_1(z) ) \approx \hat f_0(z)}$ up to quadratic errors (compare with the Newton approximation ${f(x_0+h) \approx f(x_0) + h f'(x_0)}$), and arrive at the equation

$\displaystyle \hat \psi_1(\lambda z) - \lambda \hat \psi_1(z) = \hat f_0(z). \ \ \ \ \ (5)$

This equation can be solved by Taylor series; the function ${\hat f_0}$ vanishes to second order at the origin and thus has a Taylor expansion

$\displaystyle \hat f_0(z) = \sum_{n=2}^\infty a_n z^n$

and then ${\hat \psi_1}$ has a Taylor expansion

$\displaystyle \hat \psi_1(z) = \sum_{n=2}^\infty \frac{a_n}{\lambda^n - \lambda} z^n.$

We take this as our definition of ${\hat \psi_1}$, define ${\psi_1(z) := z + \hat \psi_1(z)}$, and then define ${f_1}$ implicitly via (4).

Let us now justify that this choice works. By (3) and the generalised Cauchy integral formula, we have ${|a_n| \leq \delta_0}$ for all ${n}$; by the Diophantine assumption on ${\lambda}$, we thus have ${|\frac{a_n}{\lambda^n - \lambda}| \ll \delta_0 n^{O(1)}}$. In particular, ${\hat \psi_1}$ converges on ${B(0,1)}$, and on the disk ${B(0, (1-\eta_0/4))}$ (say) we have the bounds

$\displaystyle |\hat \psi_1(z)|, |\hat \psi'_1(z)| \ll \delta_0 \sum_{n=2}^\infty n^{O(1)} (1-\eta_0/4)^n \ll \eta_0^{-O(1)} \delta_0. \ \ \ \ \ (6)$

In particular, as ${\delta_0}$ is so small, we see that ${\psi_1}$ maps ${B(0, (1-\eta_0/4))}$ injectively to ${B(0,1)}$ and ${B(0,1-\eta_0)}$ to ${B(0,1-3\eta_0/4)}$, and the inverse ${\psi_1^{-1}}$ maps ${B(0, (1-\eta_0/2))}$ to ${B(0, (1-\eta_0/4))}$. From (3) we see that ${f_0}$ maps ${B(0,1-3\eta_0/4)}$ to ${B(0,1-\eta_0/2)}$, and so if we set ${f_1: B(0,1-\eta_0) \rightarrow B(0,1-\eta_0/4)}$ to be the function ${f_1 := \psi_1^{-1} \circ f_0 \circ \psi_1}$, then ${f_1}$ is a holomorphic function obeying (4). Expanding (4) in terms of ${\hat f_0}$ and ${\hat \psi_1}$ as before, and also writing ${f_1(z) = \lambda z + \hat f_1(z)}$, we have

$\displaystyle \lambda z + \lambda \hat \psi_1(z) + \hat f_0( z + \hat \psi_1(z) ) = \lambda z + \hat f_1(z) + \hat \psi_1(\lambda z + \hat f_1(z))$

for ${z \in B(0, 1-\eta_0)}$, which by (5) simplifies to

$\displaystyle \hat f_1(z) = \hat f_0( z + \hat \psi_1(z) ) - \hat f_0(z) + \hat \psi_1(\lambda z) - \hat \psi_1(\lambda z + \hat f_1(z)).$

From (6), the fundamental theorem of calculus, and the smallness of ${\delta_0}$ we have

$\displaystyle |\hat \psi_1(\lambda z) - \hat \psi_1(\lambda z + \hat f_1(z))| \leq \frac{1}{2} |\hat f_1(z)|$

and thus

$\displaystyle |\hat f_1(z)| \leq 2 |\hat f_0( z + \hat \psi_1(z) ) - \hat f_0(z)|.$

From (3) and the Cauchy integral formula we have ${\hat f'_0(z) = O( \delta_0 \eta_0^{-O(1)})}$ on (say) ${B(0,1-\eta_0/4)}$, and so from (6) and the fundamental theorem of calculus we conclude that

$\displaystyle |\hat f_1(z)| \ll \delta_0^2 \eta_0^{-O(1)}$

on ${B(0,1-\eta_0)}$, and the claim follows. $\Box$

If we set ${\eta_0 := 1/2}$, ${f_0 := f}$, and ${\delta_0>0}$ to be sufficiently small, then (since ${f(z)-\lambda z}$ vanishes to second order at the origin), the hypotheses of this lemma will be obeyed for some sufficiently small ${r_0}$. Iterating the lemma (and halving ${\eta_0}$ repeatedly), we can then find sequences ${\eta_n, \delta_n, r_n > 0}$, injective holomorphic functions ${\psi_n: B(0,r_n) \rightarrow B(0,r_{n-1})}$ and holomorphic functions ${f_n: B(0,r_n) \rightarrow {\bf C}}$ such that one has the recursive identities and estimates

$\displaystyle \eta_n = \eta_{n-1} / 2$

$\displaystyle r_n = (1 - \eta_{n-1}) r_{n-1}$

$\displaystyle \delta_n = O( \delta_{n-1}^2 \eta_{n-1}^{-O(1)} )$

$\displaystyle |\psi_n(z) - z| \ll \delta_{n-1} \eta_{n-1}^{-O(1)} r_n$

$\displaystyle |f_n(z) - \lambda z| \leq \delta_n r_n$

$\displaystyle f_{n-1}( \psi_n(z) ) = \psi_n(f_n(z))$

for all ${n \geq 1}$ and ${z \in B(0,r_n)}$. By construction, ${r_n}$ decreases to a positive radius ${r_\infty}$ that is a constant multiple of ${r_0}$, while (for ${\delta_0}$ small enough) ${\delta_n}$ converges double-exponentially to zero, so in particular ${f_n(z)}$ converges uniformly to ${\lambda z}$ on ${B(0,r_\infty)}$. Also, ${\psi_n}$ is close enough to the identity, the compositions ${\Psi_n := \psi_1 \circ \dots \circ \psi_n}$ are uniformly convergent on ${B(0,r_\infty/2)}$ with ${\Psi_n(0)=0}$ and ${\Psi'_n(0)=1}$. From this we have

$\displaystyle f( \Psi_n(z) ) = \Psi_n(f_n(z))$

on ${B(0,r_\infty/4)}$, and on taking limits using Morera’s theorem we obtain a holomorphic function ${\Psi}$ defined near ${0}$ with ${\Psi(0)=0}$, ${\Psi'(0)=1}$, and

$\displaystyle f( \Psi(z) ) = \Psi(\lambda z),$

obtaining the required linearisation.

Remark 5 The idea of using a Newton-type method to obtain error terms that decay double-exponentially, and can therefore absorb exponential losses in the iteration, also occurs in KAM theory and in Nash-Moser iteration, presumably due to Siegel’s influence on Moser. (I discuss Nash-Moser iteration in this note that I wrote back in 2006.)

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if ${U: H \rightarrow H}$ is a unitary operator on a Hilbert space ${H}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \pi_{H^U} v$

in the strong topology, where ${H^U := \{ w \in H: Uw = w \}}$ is the ${U}$-invariant subspace of ${H}$, and ${\pi_{H^U}}$ is the orthogonal projection to ${H^U}$. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if ${G}$ is a countable amenable group acting on a Hilbert space ${H}$ by unitary transformations ${T^g: H \rightarrow H}$ for ${g \in G}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \mathop{\bf E}_{g \in \Phi_N} T^g v = \pi_{H^G} v \ \ \ \ \ (1)$

for any Folner sequence ${\Phi_N}$ of ${G}$, where ${H^G := \{ w \in H: T^g w = w \hbox{ for all }g \in G \}}$ is the ${G}$-invariant subspace, and ${\mathop{\bf E}_{a \in A} f(a) := \frac{1}{|A|} \sum_{a \in A} f(a)}$ is the average of ${f}$ on ${A}$. Thus one can interpret ${\pi_{H^G} v}$ as a certain average of elements of the orbit ${Gv := \{ T^g v: g \in G \}}$ of ${v}$.

In a previous blog post, I noted a variant of this ergodic theorem (due to Alaoglu and Birkhoff) that holds even when the group ${G}$ is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem) Let ${G}$ be an arbitrary group acting unitarily on a Hilbert space ${H}$, and let ${v}$ be a vector in ${H}$. Then ${\pi_{H^G} v}$ is the element in the closed convex hull of ${Gv := \{ T^g v: g \in G \}}$ of minimal norm, and is also the unique element of ${H^G}$ in this closed convex hull.

I recently stumbled upon a different way to think about this theorem, in the additive case ${G = (G,+)}$ when ${G}$ is abelian, which has a closer resemblance to the classical mean ergodic theorem. Given an arbitrary additive group ${G = (G,+)}$ (not necessarily discrete, or countable), let ${{\mathcal F}}$ denote the collection of finite non-empty multisets in ${G}$ – that is to say, unordered collections ${\{a_1,\dots,a_n\}}$ of elements ${a_1,\dots,a_n}$ of ${G}$, not necessarily distinct, for some positive integer ${n}$. Given two multisets ${A = \{a_1,\dots,a_n\}}$, ${B = \{b_1,\dots,b_m\}}$ in ${{\mathcal F}}$, we can form the sum set ${A + B := \{ a_i + b_j: 1 \leq i \leq n, 1 \leq j \leq m \}}$. Note that the sum set ${A+B}$ can contain multiplicity even when ${A, B}$ do not; for instance, ${\{ 1,2\} + \{1,2\} = \{2,3,3,4\}}$. Given a multiset ${A = \{a_1,\dots,a_n\}}$ in ${{\mathcal F}}$, and a function ${f: G \rightarrow H}$ from ${G}$ to a vector space ${H}$, we define the average ${\mathop{\bf E}_{a \in A} f(a)}$ as

$\displaystyle \mathop{\bf E}_{a \in A} f(a) = \frac{1}{n} \sum_{j=1}^n f(a_j).$

Note that the multiplicity function of the set ${A}$ affects the average; for instance, we have ${\mathop{\bf E}_{a \in \{1,2\}} a = \frac{3}{2}}$, but ${\mathop{\bf E}_{a \in \{1,2,2\}} a = \frac{5}{3}}$.

We can define a directed set on ${{\mathcal F}}$ as follows: given two multisets ${A,B \in {\mathcal F}}$, we write ${A \geq B}$ if we have ${A = B+C}$ for some ${C \in {\mathcal F}}$. Thus for instance we have ${\{ 1, 2, 2, 3\} \geq \{1,2\}}$. It is easy to verify that this operation is transitive and reflexive, and is directed because any two elements ${A,B}$ of ${{\mathcal F}}$ have a common upper bound, namely ${A+B}$. (This is where we need ${G}$ to be abelian.) The notion of convergence along a net, now allows us to define the notion of convergence along ${{\mathcal F}}$; given a family ${x_A}$ of points in a topological space ${X}$ indexed by elements ${A}$ of ${{\mathcal F}}$, and a point ${x}$ in ${X}$, we say that ${x_A}$ converges to ${x}$ along ${{\mathcal F}}$ if, for every open neighbourhood ${U}$ of ${x}$ in ${X}$, one has ${x_A \in U}$ for sufficiently large ${A}$, that is to say there exists ${B \in {\mathcal F}}$ such that ${x_A \in U}$ for all ${A \geq B}$. If the topological space ${V}$ is Hausdorff, then the limit ${x}$ is unique (if it exists), and we then write

$\displaystyle x = \lim_{A \rightarrow G} x_A.$

When ${x_A}$ takes values in the reals, one can also define the limit superior or limit inferior along such nets in the obvious fashion.

We can then give an alternate formulation of the abstract ergodic theorem in the abelian case:

Theorem 2 (Abelian abstract ergodic theorem) Let ${G = (G,+)}$ be an arbitrary additive group acting unitarily on a Hilbert space ${H}$, and let ${v}$ be a vector in ${H}$. Then we have

$\displaystyle \pi_{H^G} v = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v$

in the strong topology of ${H}$.

Proof: Suppose that ${A \geq B}$, so that ${A=B+C}$ for some ${C \in {\mathcal F}}$, then

$\displaystyle \mathop{\bf E}_{a \in A} T^a v = \mathop{\bf E}_{c \in C} T^c ( \mathop{\bf E}_{b \in B} T^b v )$

so by unitarity and the triangle inequality we have

$\displaystyle \| \mathop{\bf E}_{a \in A} T^a v \|_H \leq \| \mathop{\bf E}_{b \in B} T^b v \|_H,$

thus ${\| \mathop{\bf E}_{a \in A} T^a v \|_H^2}$ is monotone non-increasing in ${A}$. Since this quantity is bounded between ${0}$ and ${\|v\|_H}$, we conclude that the limit ${\lim_{A \rightarrow G} \| \mathop{\bf E}_{a \in A} T^a v \|_H^2}$ exists. Thus, for any ${\varepsilon > 0}$, we have for sufficiently large ${A}$ that

$\displaystyle \| \mathop{\bf E}_{b \in B} T^b v \|_H^2 \geq \| \mathop{\bf E}_{a \in A} T^a v \|_H^2 - \varepsilon$

for all ${B \geq A}$. In particular, for any ${g \in G}$, we have

$\displaystyle \| \mathop{\bf E}_{b \in A + \{0,g\}} T^b v \|_H^2 \geq \| \mathop{\bf E}_{a \in A} T^a v \|_H^2 - \varepsilon.$

We can write

$\displaystyle \mathop{\bf E}_{b \in A + \{0,g\}} T^b v = \frac{1}{2} \mathop{\bf E}_{a \in A} T^a v + \frac{1}{2} T^g \mathop{\bf E}_{a \in A} T^a v$

and so from the parallelogram law and unitarity we have

$\displaystyle \| \mathop{\bf E}_{a \in A} T^a v - T^g \mathop{\bf E}_{a \in A} T^a v \|_H^2 \leq 4 \varepsilon$

for all ${g \in G}$, and hence by the triangle inequality (averaging ${g}$ over a finite multiset ${C}$)

$\displaystyle \| \mathop{\bf E}_{a \in A} T^a v - \mathop{\bf E}_{b \in A+C} T^b v \|_H^2 \leq 4 \varepsilon$

for any ${C \in {\mathcal F}}$. This shows that ${\mathop{\bf E}_{a \in A} T^a v}$ is a Cauchy sequence in ${H}$ (in the strong topology), and hence (by the completeness of ${H}$) tends to a limit. Shifting ${A}$ by a group element ${g}$, we have

$\displaystyle \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A + \{g\}} T^a v = T^g \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v$

and hence ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v}$ is invariant under shifts, and thus lies in ${H^G}$. On the other hand, for any ${w \in H^G}$ and ${A \in {\mathcal F}}$, we have

$\displaystyle \langle \mathop{\bf E}_{a \in A} T^a v, w \rangle_H = \mathop{\bf E}_{a \in A} \langle v, T^{-a} w \rangle_H = \langle v, w \rangle_H$

and thus on taking strong limits

$\displaystyle \langle \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v, w \rangle_H = \langle v, w \rangle_H$

and so ${v - \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v}$ is orthogonal to ${H^G}$. Combining these two facts we see that ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^a v}$ is equal to ${\pi_{H^G} v}$ as claimed. $\Box$

To relate this result to the classical ergodic theorem, we observe

Lemma 3 Let ${G}$ be a countable additive group, with a F{\o}lner sequence ${\Phi_n}$, and let ${f_g}$ be a bounded sequence in a normed vector space indexed by ${G}$. If ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} f_a}$ exists, then ${\lim_{n \rightarrow \infty} \mathop{\bf E}_{a \in \Phi_n} f_a}$ exists, and the two limits are equal.

Proof: From the F{\o}lner property, we see that for any ${A}$ and any ${\varepsilon>0}$, the averages ${\mathop{\bf E}_{a \in \Phi_n} f_a}$ and ${\mathop{\bf E}_{a \in A+\Phi_n} f_a}$ differ by at most ${\varepsilon}$ in norm if ${n}$ is sufficiently large depending on ${A}$, ${\varepsilon}$ (and the ${f_a}$). On the other hand, by the existence of the limit ${\lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} f_a}$, the averages ${\mathop{\bf E}_{a \in A} f_a}$ and ${\mathop{\bf E}_{a \in A + \Phi_n} f_a}$ differ by at most ${\varepsilon}$ in norm if ${A}$ is sufficiently large depending on ${\varepsilon}$ (regardless of how large ${n}$ is). The claim follows. $\Box$

It turns out that this approach can also be used as an alternate way to construct the GowersHost-Kra seminorms in ergodic theory, which has the feature that it does not explicitly require any amenability on the group ${G}$ (or separability on the underlying measure space), though, as pointed out to me in comments, even uncountable abelian groups are amenable in the sense of possessing an invariant mean, even if they do not have a F{\o}lner sequence.

Given an arbitrary additive group ${G}$, define a ${G}$-system ${({\mathrm X}, T)}$ to be a probability space ${{\mathrm X} = (X, {\mathcal X}, \mu)}$ (not necessarily separable or standard Borel), together with a collection ${T^g: X \rightarrow X}$ of invertible, measure-preserving maps, such that ${T^0}$ is the identity and ${T^g T^h = T^{g+h}}$ (modulo null sets) for all ${g,h \in G}$. This then gives isomorphisms ${T^g: L^p({\mathrm X}) \rightarrow L^p({\mathrm X})}$ for ${1 \leq p \leq \infty}$ by setting ${T^g f(x) := f(T^{-g} x)}$. From the above abstract ergodic theorem, we see that

$\displaystyle {\mathbf E}( f | {\mathcal X}^G ) = \lim_{A \rightarrow G} \mathop{\bf E}_{a \in A} T^g f$

in the strong topology of ${L^2({\mathrm X})}$ for any ${f \in L^2({\mathrm X})}$, where ${{\mathcal X}^G}$ is the collection of measurable sets ${E}$ that are essentially ${G}$-invariant in the sense that ${T^g E = E}$ modulo null sets for all ${g \in G}$, and ${{\mathbf E}(f|{\mathcal X}^G)}$ is the conditional expectation of ${f}$ with respect to ${{\mathcal X}^G}$.

In a similar spirit, we have

Theorem 4 (Convergence of Gowers-Host-Kra seminorms) Let ${({\mathrm X},T)}$ be a ${G}$-system for some additive group ${G}$. Let ${d}$ be a natural number, and for every ${\omega \in\{0,1\}^d}$, let ${f_\omega \in L^{2^d}({\mathrm X})}$, which for simplicity we take to be real-valued. Then the expression

$\displaystyle \langle (f_\omega)_{\omega \in \{0,1\}^d} \rangle_{U^d({\mathrm X})} := \lim_{A_1,\dots,A_d \rightarrow G}$

$\displaystyle \mathop{\bf E}_{h_1 \in A_1-A_1,\dots,h_d \in A_d-A_d} \int_X \prod_{\omega \in \{0,1\}^d} T^{\omega_1 h_1 + \dots + \omega_d h_d} f_\omega\ d\mu$

converges, where we write ${\omega = (\omega_1,\dots,\omega_d)}$, and we are using the product direct set on ${{\mathcal F}^d}$ to define the convergence ${A_1,\dots,A_d \rightarrow G}$. In particular, for ${f \in L^{2^d}({\mathrm X})}$, the limit

$\displaystyle \| f \|_{U^d({\mathrm X})}^{2^d} = \lim_{A_1,\dots,A_d \rightarrow G}$

$\displaystyle \mathop{\bf E}_{h_1 \in A_1-A_1,\dots,h_d \in A_d-A_d} \int_X \prod_{\omega \in \{0,1\}^d} T^{\omega_1 h_1 + \dots + \omega_d h_d} f\ d\mu$

converges.

We prove this theorem below the fold. It implies a number of other known descriptions of the Gowers-Host-Kra seminorms ${\|f\|_{U^d({\mathrm X})}}$, for instance that

$\displaystyle \| f \|_{U^d({\mathrm X})}^{2^d} = \lim_{A \rightarrow G} \mathop{\bf E}_{h \in A-A} \| f T^h f \|_{U^{d-1}({\mathrm X})}^{2^{d-1}}$

for ${d > 1}$, while from the ergodic theorem we have

$\displaystyle \| f \|_{U^1({\mathrm X})} = \| {\mathbf E}( f | {\mathcal X}^G ) \|_{L^2({\mathrm X})}.$

This definition also manifestly demonstrates the cube symmetries of the Host-Kra measures ${\mu^{[d]}}$ on ${X^{\{0,1\}^d}}$, defined via duality by requiring that

$\displaystyle \langle (f_\omega)_{\omega \in \{0,1\}^d} \rangle_{U^d({\mathrm X})} = \int_{X^{\{0,1\}^d}} \bigotimes_{\omega \in \{0,1\}^d} f_\omega\ d\mu^{[d]}.$

In a subsequent blog post I hope to present a more detailed study of the ${U^2}$ norm and its relationship with eigenfunctions and the Kronecker factor, without assuming any amenability on ${G}$ or any separability or topological structure on ${{\mathrm X}}$.

Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices ${M_n}$ of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power ${n^{-A}}$ of ${n}$. This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices ${M_n}$ (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on ${M_n}$) a result of the form

$\displaystyle {\bf P} (\lambda_{i+1}(M_n) - \lambda_i(M_n) \leq n^{-A} ) \leq n^{-B}$

for any ${B}$ and ${i}$, if ${A}$ is sufficiently large depending on ${B}$ (in a linear fashion), and ${n}$ is sufficiently large depending on ${B}$. The point here is that ${B}$ can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)

In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of ${M_n}$ have unit variance, so that the eigenvalues of ${M_n}$ are ${O(\sqrt{n})}$ with high probability), giving the bound

$\displaystyle {\bf P} (\lambda_{i+1}(M_n) - \lambda_i(M_n) \leq \delta / \sqrt{n} ) \ll \delta \ \ \ \ \ (1)$

for ${\delta \geq n^{-O(1)}}$ (one also has good results of this type for smaller values of ${\delta}$). This is only optimal in the regime ${\delta \sim 1}$; we expect to establish some eigenvalue repulsion, improving the RHS to ${\delta^2}$ for real matrices and ${\delta^3}$ for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form

$\displaystyle {\bf P} (\lambda_{i+k}(M_n) - \lambda_i(M_n) \leq \delta / \sqrt{n} ) \ll \delta^{ck^2}$

for any fixed ${k \geq 1}$ and some absolute constant ${c>0}$ (which we can asymptotically make to be ${1/3}$ for large ${k}$, though it ought to be as large as ${1}$), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.

In the case of Erdös-Renyi graphs, we don’t have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn’t quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of ${n^{o(1)}}$ on the RHS.

As a sample applications of the eigenvalue separation results, we can now obtain some information about eigenvectors; for instance, we can show that the components of the eigenvectors all have magnitude at least ${n^{-A}}$ for some ${A}$ with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)

We have seen in previous notes that the operation of forming a Dirichlet series

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

or twisted Dirichlet series

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

is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fourier transform, since the functions ${n \mapsto \frac{1}{n^s}}$ and ${n \mapsto \frac{\chi(n)}{n^s}}$ are multiplicative characters.

Similarly, it turns out that the operation of forming an additive Fourier series

$\displaystyle \hat f(\theta) := \sum_n f(n) e(-n \theta),$

where ${\theta}$ lies on the (additive) unit circle ${{\bf R}/{\bf Z}}$ and ${e(\theta) := e^{2\pi i \theta}}$ is the standard additive character, is an incredibly useful tool for additive number theory, particularly when studying additive problems involving three or more variables taking values in sets such as the primes; the deployment of this tool is generally known as the Hardy-Littlewood circle method. (In the analytic number theory literature, the minus sign in the phase ${e(-n\theta)}$ is traditionally omitted, and what is denoted by ${\hat f(\theta)}$ here would be referred to instead by ${S_f(-\theta)}$, ${S(f;-\theta)}$ or just ${S(-\theta)}$.) We list some of the most classical problems in this area:

• (Even Goldbach conjecture) Is it true that every even natural number ${N}$ greater than two can be expressed as the sum ${p_1+p_2}$ of two primes?
• (Odd Goldbach conjecture) Is it true that every odd natural number ${N}$ greater than five can be expressed as the sum ${p_1+p_2+p_3}$ of three primes?
• (Waring problem) For each natural number ${k}$, what is the least natural number ${g(k)}$ such that every natural number ${N}$ can be expressed as the sum of ${g(k)}$ or fewer ${k^{th}}$ powers?
• (Asymptotic Waring problem) For each natural number ${k}$, what is the least natural number ${G(k)}$ such that every sufficiently large natural number ${N}$ can be expressed as the sum of ${G(k)}$ or fewer ${k^{th}}$ powers?
• (Partition function problem) For any natural number ${N}$, let ${p(N)}$ denote the number of representations of ${N}$ of the form ${N = n_1 + \dots + n_k}$ where ${k}$ and ${n_1 \geq \dots \geq n_k}$ are natural numbers. What is the asymptotic behaviour of ${p(N)}$ as ${N \rightarrow \infty}$?

The Waring problem and its asymptotic version will not be discussed further here, save to note that the Vinogradov mean value theorem (Theorem 13 from Notes 5) and its variants are particularly useful for getting good bounds on ${G(k)}$; see for instance the ICM article of Wooley for recent progress on these problems. Similarly, the partition function problem was the original motivation of Hardy and Littlewood in introducing the circle method, but we will not discuss it further here; see e.g. Chapter 20 of Iwaniec-Kowalski for a treatment.

Instead, we will focus our attention on the odd Goldbach conjecture as our model problem. (The even Goldbach conjecture, which involves only two variables instead of three, is unfortunately not amenable to a circle method approach for a variety of reasons, unless the statement is replaced with something weaker, such as an averaged statement; see this previous blog post for further discussion. On the other hand, the methods here can obtain weaker versions of the even Goldbach conjecture, such as showing that “almost all” even numbers are the sum of two primes; see Exercise 34 below.) In particular, we will establish the following celebrated theorem of Vinogradov:

Theorem 1 (Vinogradov’s theorem) Every sufficiently large odd number ${N}$ is expressible as the sum of three primes.

Recently, the restriction that ${n}$ be sufficiently large was replaced by Helfgott with ${N > 5}$, thus establishing the odd Goldbach conjecture in full. This argument followed the same basic approach as Vinogradov (based on the circle method), but with various estimates replaced by “log-free” versions (analogous to the log-free zero-density theorems in Notes 7), combined with careful numerical optimisation of constants and also some numerical work on the even Goldbach problem and on the generalised Riemann hypothesis. We refer the reader to Helfgott’s text for details.

We will in fact show the more precise statement:

Theorem 2 (Quantitative Vinogradov theorem) Let ${N \geq 2}$ be an natural number. Then

$\displaystyle \sum_{a,b,c: a+b+c=N} \Lambda(a) \Lambda(b) \Lambda(c) = G_3(N) \frac{N^2}{2} + O_A( N^2 \log^{-A} N )$

for any ${A>0}$, where

$\displaystyle G_3(N) = \prod_{p|N} (1-\frac{1}{(p-1)^2}) \times \prod_{p \not | N} (1 + \frac{1}{(p-1)^3}). \ \ \ \ \ (1)$

The implied constants are ineffective.

We dropped the hypothesis that ${N}$ is odd in Theorem 2, but note that ${G_3(N)}$ vanishes when ${N}$ is even. For odd ${N}$, we have

$\displaystyle 1 \ll G_3(N) \ll 1.$

Exercise 3 Show that Theorem 2 implies Theorem 1.

Unfortunately, due to the ineffectivity of the constants in Theorem 2 (a consequence of the reliance on the Siegel-Walfisz theorem in the proof of that theorem), one cannot quantify explicitly what “sufficiently large” means in Theorem 1 directly from Theorem 2. However, there is a modification of this theorem which gives effective bounds; see Exercise 32 below.

Exercise 4 Obtain a heuristic derivation of the main term ${G_3(N) \frac{N^2}{2}}$ using the modified Cramér model (Section 1 of Supplement 4).

To prove Theorem 2, we consider the more general problem of estimating sums of the form

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c)$

for various integers ${N}$ and functions ${f,g,h: {\bf Z} \rightarrow {\bf C}}$, which we will take to be finitely supported to avoid issues of convergence.

Suppose that ${f,g,h}$ are supported on ${\{1,\dots,N\}}$; for simplicity, let us first assume the pointwise bound ${|f(n)|, |g(n)|, |h(n)| \ll 1}$ for all ${n}$. (This simple case will not cover the case in Theorem 2, when ${f,g,h}$ are truncated versions of the von Mangoldt function ${\Lambda}$, but will serve as a warmup to that case.) Then we have the trivial upper bound

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) \ll N^2. \ \ \ \ \ (2)$

A basic observation is that this upper bound is attainable if ${f,g,h}$ all “pretend” to behave like the same additive character ${n \mapsto e(\theta n)}$ for some ${\theta \in {\bf R}/{\bf Z}}$. For instance, if ${f(n)=g(n)=h(n) = e(\theta n) 1_{n \leq N}}$, then we have ${f(a)g(b)h(c) = e(\theta N)}$ when ${a+b+c=N}$, and then it is not difficult to show that

$\displaystyle \sum_{a,b,c \in {\bf Z}: a+b+c=N} f(a) g(b) h(c) = (\frac{1}{2}+o(1)) e(\theta N) N^2$

as ${N \rightarrow \infty}$.

The key to the success of the circle method lies in the converse of the above statement: the only way that the trivial upper bound (2) comes close to being sharp is when ${f,g,h}$ all correlate with the same character ${n \mapsto e(\theta n)}$, or in other words ${\hat f(\theta), \hat g(\theta), \hat h(\theta)}$ are simultaneously large. This converse is largely captured by the following two identities:

Exercise 5 Let ${f,g,h: {\bf Z} \rightarrow {\bf C}}$ be finitely supported functions. Then for any natural number ${N}$, show that

$\displaystyle \sum_{a,b,c: a+b+c=N} f(a) g(b) h(c) = \int_{{\bf R}/{\bf Z}} \hat f(\theta) \hat g(\theta) \hat h(\theta) e(\theta N)\ d\theta \ \ \ \ \ (3)$

and

$\displaystyle \sum_n |f(n)|^2 = \int_{{\bf R}/{\bf Z}} |\hat f(\theta)|^2\ d\theta.$

The traditional approach to using the circle method to compute sums such as ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$ proceeds by invoking (3) to express this sum as an integral over the unit circle, then dividing the unit circle into “major arcs” where ${\hat f(\theta), \hat g(\theta),\hat h(\theta)}$ are large but computable with high precision, and “minor arcs” where one has estimates to ensure that ${\hat f(\theta), \hat g(\theta),\hat h(\theta)}$ are small in both ${L^\infty}$ and ${L^2}$ senses. For functions ${f,g,h}$ of number-theoretic significance, such as truncated von Mangoldt functions, the “major arcs” typically consist of those ${\theta}$ that are close to a rational number ${\frac{a}{q}}$ with ${q}$ not too large, and the “minor arcs” consist of the remaining portions of the circle. One then obtains lower bounds on the contributions of the major arcs, and upper bounds on the contribution of the minor arcs, in order to get good lower bounds on ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$.

This traditional approach is covered in many places, such as this text of Vaughan. We will emphasise in this set of notes a slightly different perspective on the circle method, coming from recent developments in additive combinatorics; this approach does not quite give the sharpest quantitative estimates, but it allows for easier generalisation to more combinatorial contexts, for instance when replacing the primes by dense subsets of the primes, or replacing the equation ${a+b+c=N}$ with some other equation or system of equations.

From Exercise 5 and Hölder’s inequality, we immediately obtain

Corollary 6 Let ${f,g,h: {\bf Z} \rightarrow {\bf C}}$ be finitely supported functions. Then for any natural number ${N}$, we have

$\displaystyle |\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)| \leq (\sum_n |f(n)|^2)^{1/2} (\sum_n |g(n)|^2)^{1/2}$

$\displaystyle \times \sup_\theta |\sum_n h(n) e(n\theta)|.$

Similarly for permutations of the ${f,g,h}$.

In the case when ${f,g,h}$ are supported on ${[1,N]}$ and bounded by ${O(1)}$, this corollary tells us that we have ${\sum_{a,b,c: a+b+c=N} f(a) g(b) h(c)}$ is ${o(N^2)}$ whenever one has ${\sum_n h(n) e(n\theta) = o(N)}$ uniformly in ${\theta}$, and similarly for permutations of ${f,g,h}$. From this and the triangle inequality, we obtain the following conclusion: if ${f}$ is supported on ${[1,N]}$ and bounded by ${O(1)}$, and ${f}$ is Fourier-approximated by another function ${g}$ supported on ${[1,N]}$ and bounded by ${O(1)}$ in the sense that

$\displaystyle \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)$

uniformly in ${\theta}$, then we have

$\displaystyle \sum_{a,b,c: a+b+c=N} f(a) f(b) f(c) = \sum_{a,b,c: a+b+c=N} g(a) g(b) g(c) + o(N^2). \ \ \ \ \ (4)$

Thus, one possible strategy for estimating the sum ${\sum_{a,b,c: a+b+c=N} f(a) f(b) f(c)}$ is, one can effectively replace (or “model”) ${f}$ by a simpler function ${g}$ which Fourier-approximates ${g}$ in the sense that the exponential sums ${\sum_n f(n) e(n\theta), \sum_n g(n) e(n\theta)}$ agree up to error ${o(N)}$. For instance:

Exercise 7 Let ${N}$ be a natural number, and let ${A}$ be a random subset of ${\{1,\dots,N\}}$, chosen so that each ${n \in \{1,\dots,N\}}$ has an independent probability of ${1/2}$ of lying in ${A}$.

• (i) If ${f := 1_A}$ and ${g := \frac{1}{2} 1_{[1,N]}}$, show that with probability ${1-o(1)}$ as ${N \rightarrow \infty}$, one has ${\sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(N)}$ uniformly in ${\theta}$. (Hint: for any fixed ${\theta}$, this can be accomplished with quite a good probability (e.g. ${1-o(N^{-2})}$) using a concentration of measure inequality, such as Hoeffding’s inequality. To obtain the uniformity in ${\theta}$, round ${\theta}$ to the nearest multiple of (say) ${1/N^2}$ and apply the union bound).
• (ii) Show that with probability ${1-o(1)}$, one has ${(\frac{1}{16}+o(1))N^2}$ representations of the form ${N=a+b+c}$ with ${a,b,c \in A}$ (with ${(a,b,c)}$ treated as an ordered triple, rather than an unordered one).

In the case when ${f}$ is something like the truncated von Mangoldt function ${\Lambda(n) 1_{n \leq N}}$, the quantity ${\sum_n |f(n)|^2}$ is of size ${O( N \log N)}$ rather than ${O( N )}$. This costs us a logarithmic factor in the above analysis, however we can still conclude that we have the approximation (4) whenever ${g}$ is another sequence with ${\sum_n |g(n)|^2 \ll N \log N}$ such that one has the improved Fourier approximation

$\displaystyle \sum_n f(n) e(n\theta) = \sum_n g(n) e(n\theta) + o(\frac{N}{\log N}) \ \ \ \ \ (5)$

uniformly in ${\theta}$. (Later on we will obtain a “log-free” version of this implication in which one does not need to gain a factor of ${\frac{1}{\log N}}$ in the error term.)

This suggests a strategy for proving Vinogradov’s theorem: find an approximant ${g}$ to some suitable truncation ${f}$ of the von Mangoldt function (e.g. ${f(n) = \Lambda(n) 1_{n \leq N}}$ or ${f(n) = \Lambda(n) \eta(n/N)}$) which obeys the Fourier approximation property (5), and such that the expression ${\sum_{a+b+c=N} g(a) g(b) g(c)}$ is easily computable. It turns out that there are a number of good options for such an approximant ${g}$. One of the quickest ways to obtain such an approximation (which is used in Chapter 19 of Iwaniec and Kowalski) is to start with the standard identity ${\Lambda = -\mu L * 1}$, that is to say

$\displaystyle \Lambda(n) = - \sum_{d|n} \mu(d) \log d,$

and obtain an approximation by truncating ${d}$ to be less than some threshold ${R}$ (which, in practice, would be a small power of ${N}$):

$\displaystyle \Lambda(n) \approx - \sum_{d \leq R: d|n} \mu(d) \log d. \ \ \ \ \ (6)$

Thus, for instance, if ${f(n) = \Lambda(n) 1_{n \leq N}}$, the approximant ${g}$ would be taken to be

$\displaystyle g(n) := - \sum_{d \leq R: d|n} \mu(d) \log d 1_{n \leq N}.$

One could also use the slightly smoother approximation

$\displaystyle \Lambda(n) \approx \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} \ \ \ \ \ (7)$

in which case we would take

$\displaystyle g(n) := \sum_{d \leq R: d|n} \mu(d) \log \frac{R}{d} 1_{n \leq N}.$

The function ${g}$ is somewhat similar to the continuous Selberg sieve weights studied in Notes 4, with the main difference being that we did not square the divisor sum as we will not need to take ${g}$ to be non-negative. As long as ${z}$ is not too large, one can use some sieve-like computations to compute expressions like ${\sum_{a+b+c=N} g(a)g(b)g(c)}$ quite accurately. The approximation (5) can be justified by using a nice estimate of Davenport that exemplifies the Mobius pseudorandomness heuristic from Supplement 4:

Theorem 8 (Davenport’s estimate) For any ${A>0}$ and ${x \geq 2}$, we have

$\displaystyle \sum_{n \leq x} \mu(n) e(\theta n) \ll_A x \log^{-A} x$

uniformly for all ${\theta \in {\bf R}/{\bf Z}}$. The implied constants are ineffective.

This estimate will be proven by splitting into two cases. In the “major arc” case when ${\theta}$ is close to a rational ${a/q}$ with ${q}$ small (of size ${O(\log^{O(1)} x)}$ or so), this estimate will be a consequence of the Siegel-Walfisz theorem ( from Notes 2); it is the application of this theorem that is responsible for the ineffective constants. In the remaining “minor arc” case, one proceeds by using a combinatorial identity (such as Vaughan’s identity) to express the sum ${\sum_{n \leq x} \mu(n) e(\theta n)}$ in terms of bilinear sums of the form ${\sum_n \sum_m a_n b_m e(\theta nm)}$, and use the Cauchy-Schwarz inequality and the minor arc nature of ${\theta}$ to obtain a gain in this case. This will all be done below the fold. We will also use (a rigorous version of) the approximation (6) (or (7)) to establish Vinogradov’s theorem.

A somewhat different looking approximation for the von Mangoldt function that also turns out to be quite useful is

$\displaystyle \Lambda(n) \approx \sum_{q \leq Q} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} ) \ \ \ \ \ (8)$

for some ${Q}$ that is not too large compared to ${N}$. The methods used to establish Theorem 8 can also establish a Fourier approximation that makes (8) precise, and which can yield an alternate proof of Vinogradov’s theorem; this will be done below the fold.

The approximation (8) can be written in a way that makes it more similar to (7):

Exercise 9 Show that the right-hand side of (8) can be rewritten as

$\displaystyle \sum_{d \leq Q: d|n} \mu(d) \rho_d$

where

$\displaystyle \rho_d := \frac{d}{\phi(d)} \sum_{m \leq Q/d: (m,d)=1} \frac{\mu^2(m)}{\phi(m)}.$

Then, show the inequalities

$\displaystyle \sum_{m \leq Q/d} \frac{\mu^2(m)}{\phi(m)} \leq \rho_d \leq \sum_{m \leq Q} \frac{\mu^2(m)}{\phi(m)}$

and conclude that

$\displaystyle \log \frac{Q}{d} - O(1) \leq \rho_d \leq \log Q + O(1).$

(Hint: for the latter estimate, use Theorem 27 of Notes 1.)

The coefficients ${\rho_d}$ in the above exercise are quite similar to optimised Selberg sieve coefficients (see Section 2 of Notes 4).

Another approximation to ${\Lambda}$, related to the modified Cramér random model (see Model 10 of Supplement 4) is

$\displaystyle \Lambda(n) \approx \frac{W}{\phi(W)} 1_{(n,W)=1} \ \ \ \ \ (9)$

where ${W := \prod_{p \leq w} p}$ and ${w}$ is a slowly growing function of ${N}$ (e.g. ${w = \log\log N}$); a closely related approximation is

$\displaystyle \frac{\phi(W)}{W} \Lambda(Wn+b) \approx 1 \ \ \ \ \ (10)$

for ${W,w}$ as above and ${1 \leq b \leq W}$ coprime to ${W}$. These approximations (closely related to a device known as the “${W}$-trick”) are not as quantitatively accurate as the previous approximations, but can still suffice to establish Vinogradov’s theorem, and also to count many other linear patterns in the primes or subsets of the primes (particularly if one injects some additional tools from additive combinatorics, and specifically the inverse conjecture for the Gowers uniformity norms); see this paper of Ben Green and myself for more discussion (and this more recent paper of Shao for an analysis of this approach in the context of Vinogradov-type theorems). The following exercise expresses the approximation (9) in a form similar to the previous approximation (8):

Exercise 10 With ${W}$ as above, show that

$\displaystyle \frac{W}{\phi(W)} 1_{(n,W)=1} = \sum_{q|W} \sum_{a \in ({\bf Z}/q{\bf Z})^\times} \frac{\mu(q)}{\phi(q)} e( \frac{an}{q} )$

for all natural numbers ${n}$.

Kaisa Matomaki, Maksym Radziwill, and I have just uploaded to the arXiv our paper “An averaged form of Chowla’s conjecture“. This paper concerns a weaker variant of the famous conjecture of Chowla (discussed for instance in this previous post) that

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

as ${X \rightarrow \infty}$ for any distinct natural numbers ${h_1,\dots,h_k}$, where ${\lambda}$ denotes the Liouville function. (One could also replace the Liouville function here by the Möbius function ${\mu}$ and obtain a morally equivalent conjecture.) This conjecture remains open for any ${k \geq 2}$; for instance the assertion

$\displaystyle \sum_{n \leq X} \lambda(n) \lambda(n+2) = o(X)$

is a variant of the twin prime conjecture (though possibly a tiny bit easier to prove), and is subject to the notorious parity barrier (as discussed in this previous post).

Our main result asserts, roughly speaking, that Chowla’s conjecture can be established unconditionally provided one has non-trivial averaging in the ${h_1,\dots,h_k}$ parameters. More precisely, one has

Theorem 1 (Chowla on the average) Suppose ${H = H(X) \leq X}$ is a quantity that goes to infinity as ${X \rightarrow \infty}$ (but it can go to infinity arbitrarily slowly). Then for any fixed ${k \geq 1}$, we have

$\displaystyle \sum_{h_1,\dots,h_k \leq H} |\sum_{n \leq X} \lambda(n+h_1) \dots \lambda(n+h_k)| = o( H^k X ).$

In fact, we can remove one of the averaging parameters and obtain

$\displaystyle \sum_{h_2,\dots,h_k \leq H} |\sum_{n \leq X} \lambda(n) \lambda(n+h_2) \dots \lambda(n+h_k)| = o( H^{k-1} X ).$

Actually we can make the decay rate a bit more quantitative, gaining about ${\frac{\log\log H}{\log H}}$ over the trivial bound. The key case is ${k=2}$; while the unaveraged Chowla conjecture becomes more difficult as ${k}$ increases, the averaged Chowla conjecture does not increase in difficulty due to the increasing amount of averaging for larger ${k}$, and we end up deducing the higher ${k}$ case of the conjecture from the ${k=2}$ case by an elementary argument.

The proof of the theorem proceeds as follows. By exploiting the Fourier-analytic identity

$\displaystyle \int_{{\mathbf T}} (\int_{\mathbf R} |\sum_{x \leq n \leq x+H} f(n) e(\alpha n)|^2 dx)^2\ d\alpha$

$\displaystyle = \sum_{|h| \leq H} (H-|h|)^2 |\sum_n f(n) \overline{f}(n+h)|^2$

(related to a standard Fourier-analytic identity for the Gowers ${U^2}$ norm) it turns out that the ${k=2}$ case of the above theorem can basically be derived from an estimate of the form

$\displaystyle \int_0^X |\sum_{x \leq n \leq x+H} \lambda(n) e(\alpha n)|\ dx = o( H X )$

uniformly for all ${\alpha \in {\mathbf T}}$. For “major arc” ${\alpha}$, close to a rational ${a/q}$ for small ${q}$, we can establish this bound from a generalisation of a recent result of Matomaki and Radziwill (discussed in this previous post) on averages of multiplicative functions in short intervals. For “minor arc” ${\alpha}$, we can proceed instead from an argument of Katai and Bourgain-Sarnak-Ziegler (discussed in this previous post).

The argument also extends to other bounded multiplicative functions than the Liouville function. Chowla’s conjecture was generalised by Elliott, who roughly speaking conjectured that the ${k}$ copies of ${\lambda}$ in Chowla’s conjecture could be replaced by arbitrary bounded multiplicative functions ${g_1,\dots,g_k}$ as long as these functions were far from a twisted Dirichlet character ${n \mapsto \chi(n) n^{it}}$ in the sense that

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

(This type of distance is incidentally now a fundamental notion in the Granville-Soundararajan “pretentious” approach to multiplicative number theory.) During our work on this project, we found that Elliott’s conjecture is not quite true as stated due to a technicality: one can cook up a bounded multiplicative function ${g}$ which behaves like ${n^{it_j}}$ on scales ${n \sim N_j}$ for some ${N_j}$ going to infinity and some slowly varying ${t_j}$, and such a function will be far from any fixed Dirichlet character whilst still having many large correlations (e.g. the pair correlations ${\sum_{n \leq N_j} g(n+1) \overline{g(n)}}$ will be large). In our paper we propose a technical “fix” to Elliott’s conjecture (replacing (1) by a truncated variant), and show that this repaired version of Elliott’s conjecture is true on the average in much the same way that Chowla’s conjecture is. (If one restricts attention to real-valued multiplicative functions, then this technical issue does not show up, basically because one can assume without loss of generality that ${t=0}$ in this case; we discuss this fact in an appendix to the paper.)

A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function ${\zeta}$ in the critical strip ${\{ \sigma+it: 0 < \sigma < 1; t \in {\bf R} \}}$ in the limit ${t \rightarrow +\infty}$. For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude ${\log |\zeta|: {\bf C} \rightarrow [-\infty,+\infty]}$ of the zeta function. (In principle, one can reconstruct a branch of ${\log \zeta}$, and hence ${\zeta}$ itself, from ${\log |\zeta|}$ using the Cauchy-Riemann equations, or tools such as the Borel-Carathéodory theorem, see Exercise 40 of Supplement 2.)

One has the classical estimate

$\displaystyle \zeta(\sigma+it) = O( t^{O(1)} )$

when ${\sigma = O(1)}$ and ${t \geq 10}$ (say), so that

$\displaystyle \log |\zeta(\sigma+it)| \leq O( \log t ). \ \ \ \ \ (1)$

(See e.g. Exercise 37 from Supplement 3.) In view of this, let us define the normalised log-magnitudes ${F_T: {\bf C} \rightarrow [-\infty,+\infty]}$ for any ${T \geq 10}$ by the formula

$\displaystyle F_T( \sigma + it ) := \frac{1}{\log T} \log |\zeta( \sigma + i(T + t) )|;$

informally, this is a normalised window into ${\log |\zeta|}$ near ${iT}$. One can rephrase several assertions about the zeta function in terms of the asymptotic behaviour of ${F_T}$. For instance:

• (i) The bound (1) implies that ${F_T}$ is asymptotically locally bounded from above in the limit ${T \rightarrow \infty}$, thus for any compact set ${K \subset {\bf C}}$ we have ${F_T(\sigma+it) \leq O_K(1)}$ for ${\sigma+it \in K}$ and ${T}$ sufficiently large. In fact the implied constant in ${K}$ only depends on the projection of ${K}$ to the real axis.
• (ii) For ${\sigma > 1}$, we have the bounds

$\displaystyle |\zeta(\sigma+it)|, \frac{1}{|\zeta(\sigma+it)|} \leq \zeta(\sigma)$

which implies that ${F_T}$ converges locally uniformly as ${T \rightarrow +\infty}$ to zero in the region ${\{ \sigma+it: \sigma > 1, t \in {\bf R} \}}$.

• (iii) The functional equation, together with the symmetry ${\zeta(\sigma-it) = \overline{\zeta(\sigma+it)}}$, implies that

$\displaystyle |\zeta(\sigma+it)| = 2^\sigma \pi^{\sigma-1} |\sin \frac{\pi(\sigma+it)}{2}| |\Gamma(1-\sigma-it)| |\zeta(1-\sigma+it)|$

which by Exercise 17 of Supplement 3 shows that

$\displaystyle F_T( 1-\sigma+it ) = \frac{1}{2}-\sigma + F_T(\sigma+it) + o(1)$

as ${T \rightarrow \infty}$, locally uniformly in ${\sigma+it}$. In particular, when combined with the previous item, we see that ${F_T(\sigma+it)}$ converges locally uniformly as ${T \rightarrow +\infty}$ to ${\frac{1}{2}-\sigma}$ in the region ${\{ \sigma+it: \sigma < 0, t \in {\bf R}\}}$.

• (iv) From Jensen’s formula (Theorem 16 of Supplement 2) we see that ${\log|\zeta|}$ is a subharmonic function, and thus ${F_T}$ is subharmonic as well. In particular we have the mean value inequality

$\displaystyle F_T( z_0 ) \leq \frac{1}{\pi r^2} \int_{z: |z-z_0| \leq r} F_T(z)$

for any disk ${\{ z: |z-z_0| \leq r \}}$, where the integral is with respect to area measure. From this and (ii) we conclude that

$\displaystyle \int_{z: |z-z_0| \leq r} F_T(z) \geq O_{z_0,r}(1)$

for any disk with ${\hbox{Re}(z_0)>1}$ and sufficiently large ${T}$; combining this with (i) we conclude that ${F_T}$ is asymptotically locally bounded in ${L^1}$ in the limit ${T \rightarrow \infty}$, thus for any compact set ${K \subset {\bf C}}$ we have ${\int_K |F_T| \ll_K 1}$ for sufficiently large ${T}$.

From (v) and the usual Arzela-Ascoli diagonalisation argument, we see that the ${F_T}$ are asymptotically compact in the topology of distributions: given any sequence ${T_n}$ tending to ${+\infty}$, one can extract a subsequence such that the ${F_T}$ converge in the sense of distributions. Let us then define a normalised limit profile of ${\log|\zeta|}$ to be a distributional limit ${F}$ of a sequence of ${F_T}$; they are analogous to limiting profiles in PDE, and also to the more recent introduction of “graphons” in the theory of graph limits. Then by taking limits in (i)-(iv) we can say a lot about such normalised limit profiles ${F}$ (up to almost everywhere equivalence, which is an issue we will address shortly):

• (i) ${F}$ is bounded from above in the critical strip ${\{ \sigma+it: 0 \leq \sigma \leq 1 \}}$.
• (ii) ${F}$ vanishes on ${\{ \sigma+it: \sigma \geq 1\}}$.
• (iii) We have the functional equation ${F(1-\sigma+it) = \frac{1}{2}-\sigma + F(\sigma+it)}$ for all ${\sigma+it}$. In particular ${F(\sigma+it) = \frac{1}{2}-\sigma}$ for ${\sigma<0}$.
• (iv) ${F}$ is subharmonic.

Unfortunately, (i)-(iv) fail to characterise ${F}$ completely. For instance, one could have ${F(\sigma+it) = f(\sigma)}$ for any convex function ${f(\sigma)}$ of ${\sigma}$ that equals ${0}$ for ${\sigma \geq 1}$, ${\frac{1}{2}-\sigma}$ for ${\sigma \leq 1}$, and obeys the functional equation ${f(1-\sigma) = \frac{1}{2}-\sigma+f(\sigma)}$, and this would be consistent with (i)-(iv). One can also perturb such examples in a region where ${f}$ is strictly convex to create further examples of functions obeying (i)-(iv). Note from subharmonicity that the function ${\sigma \mapsto \sup_t F(\sigma+it)}$ is always going to be convex in ${\sigma}$; this can be seen as a limiting case of the Hadamard three-lines theorem (Exercise 41 of Supplement 2).

We pause to address one minor technicality. We have defined ${F}$ as a distributional limit, and as such it is a priori only defined up to almost everywhere equivalence. However, due to subharmonicity, there is a unique upper semi-continuous representative of ${F}$ (taking values in ${[-\infty,+\infty)}$), defined by the formula

$\displaystyle F(z_0) = \lim_{r \rightarrow 0^+} \frac{1}{\pi r^2} \int_{B(z_0,r)} F(z)\ dz$

for any ${z_0 \in {\bf C}}$ (note from subharmonicity that the expression in the limit is monotone nonincreasing as ${r \rightarrow 0}$, and is also continuous in ${z_0}$). We will now view this upper semi-continuous representative of ${F}$ as the canonical representative of ${F}$, so that ${F}$ is now defined everywhere, rather than up to almost everywhere equivalence.

By a classical theorem of Riesz, a function ${F}$ is subharmonic if and only if the distribution ${-\Delta F}$ is a non-negative measure, where ${\Delta := \frac{\partial^2}{\partial \sigma^2} + \frac{\partial^2}{\partial t^2}}$ is the Laplacian in the ${\sigma,t}$ coordinates. Jensen’s formula (or Greens’ theorem), when interpreted distributionally, tells us that

$\displaystyle -\Delta \log |\zeta| = \frac{1}{2\pi} \sum_\rho \delta_\rho$

away from the real axis, where ${\rho}$ ranges over the non-trivial zeroes of ${\zeta}$. Thus, if ${F}$ is a normalised limit profile for ${\log |\zeta|}$ that is the distributional limit of ${F_{T_n}}$, then we have

$\displaystyle -\Delta F = \nu$

where ${\nu}$ is a non-negative measure which is the limit in the vague topology of the measures

$\displaystyle \nu_{T_n} := \frac{1}{2\pi \log T_n} \sum_\rho \delta_{\rho - T_n}.$

Thus ${\nu}$ is a normalised limit profile of the zeroes of the Riemann zeta function.

Using this machinery, we can recover many classical theorems about the Riemann zeta function by “soft” arguments that do not require extensive calculation. Here are some examples:

Theorem 1 The Riemann hypothesis implies the Lindelöf hypothesis.

Proof: It suffices to show that any limiting profile ${F}$ (arising as the limit of some ${F_{T_n}}$) vanishes on the critical line ${\{1/2+it: t \in {\bf R}\}}$. But if the Riemann hypothesis holds, then the measures ${\nu_{T_n}}$ are supported on the critical line ${\{1/2+it: t \in {\bf R}\}}$, so the normalised limit profile ${\nu}$ is also supported on this line. This implies that ${F}$ is harmonic outside of the critical line. By (ii) and unique continuation for harmonic functions, this implies that ${F}$ vanishes on the half-space ${\{ \sigma+it: \sigma \geq \frac{1}{2} \}}$ (and equals ${\frac{1}{2}-\sigma}$ on the complementary half-space, by (iii)), giving the claim. $\Box$

In fact, we have the following sharper statement:

Theorem 2 (Backlund) The Lindelöf hypothesis is equivalent to the assertion that for any fixed ${\sigma_0 > \frac{1}{2}}$, the number of zeroes in the region ${\{ \sigma+it: \sigma > \sigma_0, T \leq t \leq T+1 \}}$ is ${o(\log T)}$ as ${T \rightarrow \infty}$.

Proof: If the latter claim holds, then for any ${T_n \rightarrow \infty}$, the measures ${\nu_{T_n}}$ assign a mass of ${o(1)}$ to any region of the form ${\{ \sigma+it: \sigma > \sigma_0; t_0 \leq t \leq t_0+1 \}}$ as ${n \rightarrow \infty}$ for any fixed ${\sigma_0>\frac{1}{2}}$ and ${t_0 \in {\bf R}}$. Thus the normalised limiting profile measure ${\nu}$ is supported on the critical line, and we can repeat the previous argument.

Conversely, suppose the claim fails, then we can find a sequence ${T_n}$ and ${\sigma_0>0}$ such that ${\nu_{T_n}}$ assigns a mass of ${\gg 1}$ to the region ${\{ \sigma+it: \sigma > \sigma_0; 0\leq t \leq 1 \}}$. Extracting a normalised limiting profile, we conclude that the normalised limiting profile measure ${\nu}$ is non-trivial somewhere to the right of the critical line, so the associated subharmonic function ${F}$ is not harmonic everywhere to the right of the critical line. From the maximum principle and (ii) this implies that ${F}$ has to be positive somewhere on the critical line, but this contradicts the Lindelöf hypothesis. (One has to take a bit of care in the last step since ${F_{T_n}}$ only converges to ${F}$ in the sense of distributions, but it turns out that the subharmonicity of all the functions involved gives enough regularity to justify the argument; we omit the details here.) $\Box$

Theorem 3 (Littlewood) Assume the Lindelöf hypothesis. Then for any fixed ${\alpha>0}$, the number of zeroes in the region ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ is ${(2\pi \alpha+o(1)) \log T}$ as ${T \rightarrow +\infty}$.

Proof: By the previous arguments, the only possible normalised limiting profile for ${\log |\zeta|}$ is ${\max( 0, \frac{1}{2}-\sigma )}$. Taking distributional Laplacians, we see that the only possible normalised limiting profile for the zeroes is Lebesgue measure on the critical line. Thus, ${\nu_T( \{\sigma+it: T \leq t \leq T+\alpha \} )}$ can only converge to ${\alpha}$ as ${T \rightarrow +\infty}$, and the claim follows. $\Box$

Even without the Lindelöf hypothesis, we have the following result:

Theorem 4 (Titchmarsh) For any fixed ${\alpha>0}$, there are ${\gg_\alpha \log T}$ zeroes in the region ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ for sufficiently large ${T}$.

Among other things, this theorem recovers a classical result of Littlewood that the gaps between the imaginary parts of the zeroes goes to zero, even without assuming unproven conjectures such as the Riemann or Lindelöf hypotheses.

Proof: Suppose for contradiction that this were not the case, then we can find ${\alpha > 0}$ and a sequence ${T_n \rightarrow \infty}$ such that ${\{ \sigma+it: T_n \leq t \leq T_n+\alpha \}}$ contains ${o(\log T)}$ zeroes. Passing to a subsequence to extract a limit profile, we conclude that the normalised limit profile measure ${\nu}$ assigns no mass to the horizontal strip ${\{ \sigma+it: 0 \leq t \leq\alpha \}}$. Thus the associated subharmonic function ${F}$ is actually harmonic on this strip. But by (ii) and unique continuation this forces ${F}$ to vanish on this strip, contradicting the functional equation (iii). $\Box$

Exercise 5 Use limiting profiles to obtain the matching upper bound of ${O_\alpha(\log T)}$ for the number of zeroes in ${\{ \sigma+it: T \leq t \leq T+\alpha \}}$ for sufficiently large ${T}$.

Remark 6 One can remove the need to take limiting profiles in the above arguments if one can come up with quantitative (or “hard”) substitutes for qualitative (or “soft”) results such as the unique continuation property for harmonic functions. This would also allow one to replace the qualitative decay rates ${o(1)}$ with more quantitative decay rates such as ${1/\log \log T}$ or ${1/\log\log\log T}$. Indeed, the classical proofs of the above theorems come with quantitative bounds that are typically of this form (see e.g. the text of Titchmarsh for details).

Exercise 7 Let ${S(T)}$ denote the quantity ${S(T) := \frac{1}{\pi} \hbox{arg} \zeta(\frac{1}{2}+iT)}$, where the branch of the argument is taken by using a line segment connecting ${\frac{1}{2}+iT}$ to (say) ${2+iT}$, and then to ${2}$. If we have a sequence ${T_n \rightarrow \infty}$ producing normalised limit profiles ${F, \nu}$ for ${\log|\zeta|}$ and the zeroes respectively, show that ${t \mapsto \frac{1}{\log T_n} S(T_n + t)}$ converges in the sense of distributions to the function ${t \mapsto \frac{1}{\pi} \int_{1/2}^1 \frac{\partial F}{\partial t}(\sigma+it)\ d\sigma}$, or equivalently

$\displaystyle t \mapsto \frac{1}{2\pi} \frac{\partial}{\partial t} \int_0^1 F(\sigma+it)\ d\sigma.$

Conclude in particular that if the Lindelöf hypothesis holds, then ${S(T) = o(\log T)}$ as ${T \rightarrow \infty}$.

A little bit more about the normalised limit profiles ${F}$ are known unconditionally, beyond (i)-(iv). For instance, from Exercise 3 of Notes 5 we have ${\zeta(1/2 + it ) = O( t^{1/6+o(1)} )}$ as ${t \rightarrow +\infty}$, which implies that any normalised limit profile ${F}$ for ${\log|\zeta|}$ is bounded by ${1/6}$ on the critical line, beating the bound of ${1/4}$ coming from convexity and (ii), (iii), and then convexity can be used to further bound ${F}$ away from the critical line also. Some further small improvements of this type are known (coming from various methods for estimating exponential sums), though they fall well short of determining ${F}$ completely at our current level of understanding. Of course, given that we believe the Riemann hypothesis (and hence the Lindelöf hypothesis) to be true, the only actual limit profile that should exist is ${\max(0,\frac{1}{2}-\sigma)}$ (in fact this assertion is equivalent to the Lindelöf hypothesis, by the arguments above).

Better control on limiting profiles is available if we do not insist on controlling ${\zeta}$ for all values of the height parameter ${T}$, but only for most such values, thanks to the existence of several mean value theorems for the zeta function, as discussed in Notes 6; we discuss this below the fold.

In analytic number theory, it is a well-known phenomenon that for many arithmetic functions ${f: {\bf N} \rightarrow {\bf C}}$ of interest in number theory, it is significazintly easier to estimate logarithmic sums such as

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

than it is to estimate summatory functions such as

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

(Here we are normalising ${f}$ to be roughly constant in size, e.g. ${f(n) = O( n^{o(1)} )}$ as ${n \rightarrow \infty}$.) For instance, when ${f}$ is the von Mangoldt function ${\Lambda}$, the logarithmic sums ${\sum_{n \leq x} \frac{\Lambda(n)}{n}}$ can be adequately estimated by Mertens’ theorem, which can be easily proven by elementary means (see Notes 1); but a satisfactory estimate on the summatory function ${\sum_{n \leq x} \Lambda(n)}$ requires the prime number theorem, which is substantially harder to prove (see Notes 2). (From a complex-analytic or Fourier-analytic viewpoint, the problem is that the logarithmic sums ${\sum_{n \leq x} \frac{f(n)}{n}}$ can usually be controlled just from knowledge of the Dirichlet series ${\sum_n \frac{f(n)}{n^s}}$ for ${s}$ near ${1}$; but the summatory functions require control of the Dirichlet series ${\sum_n \frac{f(n)}{n^s}}$ for ${s}$ on or near a large portion of the line ${\{ 1+it: t \in {\bf R} \}}$. See Notes 2 for further discussion.)

Viewed conversely, whenever one has a difficult estimate on a summatory function such as ${\sum_{n \leq x} f(n)}$, one can look to see if there is a “cheaper” version of that estimate that only controls the logarithmic sums ${\sum_{n \leq x} \frac{f(n)}{n}}$, which is easier to prove than the original, more “expensive” estimate. In this post, we shall do this for two theorems, a classical theorem of Halasz on mean values of multiplicative functions on long intervals, and a much more recent result of Matomaki and Radziwiłł on mean values of multiplicative functions in short intervals. The two are related; the former theorem is an ingredient in the latter (though in the special case of the Matomaki-Radziwiłł theorem considered here, we will not need Halasz’s theorem directly, instead using a key tool in the proof of that theorem).

We begin with Halasz’s theorem. Here is a version of this theorem, due to Montgomery and to Tenenbaum:

Theorem 1 (Halasz-Montgomery-Tenenbaum) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a multiplicative function with ${|f(n)| \leq 1}$ for all ${n}$. Let ${x \geq 3}$ and ${T \geq 1}$, and set

$\displaystyle M := \min_{|t| \leq T} \sum_{p \leq x} \frac{1 - \hbox{Re}( f(p) p^{-it} )}{p}.$

Then one has

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

Informally, this theorem asserts that ${\sum_{n \leq x} f(n)}$ is small compared with ${x}$, unless ${f}$ “pretends” to be like the character ${p \mapsto p^{it}}$ on primes for some small ${y}$. (This is the starting point of the “pretentious” approach of Granville and Soundararajan to analytic number theory, as developed for instance here.) We now give a “cheap” version of this theorem which is significantly weaker (both because it settles for controlling logarithmic sums rather than summatory functions, it requires ${f}$ to be completely multiplicative instead of multiplicative, it requires a strong bound on the analogue of the quantity ${M}$, and because it only gives qualitative decay rather than quantitative estimates), but easier to prove:

Theorem 2 (Cheap Halasz) Let ${x}$ be an asymptotic parameter goingto infinity. Let ${f: {\bf N} \rightarrow {\bf C}}$ be a completely multiplicative function (possibly depending on ${x}$) such that ${|f(n)| \leq 1}$ for all ${n}$, such that

$\displaystyle \sum_{p \leq x} \frac{1 - \hbox{Re}( f(p) )}{p} \gg \log\log x. \ \ \ \ \ (1)$

Then

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

Note that now that we are content with estimating exponential sums, we no longer need to preclude the possibility that ${f(p)}$ pretends to be like ${p^{it}}$; see Exercise 11 of Notes 1 for a related observation.

To prove this theorem, we first need a special case of the Turan-Kubilius inequality.

Lemma 3 (Turan-Kubilius) Let ${x}$ be a parameter going to infinity, and let ${1 < P < x}$ be a quantity depending on ${x}$ such that ${P = x^{o(1)}}$ and ${P \rightarrow \infty}$ as ${x \rightarrow \infty}$. Then

$\displaystyle \sum_{n \leq x} \frac{ | \frac{1}{\log \log P} \sum_{p \leq P: p|n} 1 - 1 |}{n} = o( \log x ).$

Informally, this lemma is asserting that

$\displaystyle \sum_{p \leq P: p|n} 1 \approx \log \log P$

for most large numbers ${n}$. Another way of writing this heuristically is in terms of Dirichlet convolutions:

$\displaystyle 1 \approx 1 * \frac{1}{\log\log P} 1_{{\mathcal P} \cap [1,P]}.$

This type of estimate was previously discussed as a tool to establish a criterion of Katai and Bourgain-Sarnak-Ziegler for Möbius orthogonality estimates in this previous blog post. See also Section 5 of Notes 1 for some similar computations.

Proof: By Cauchy-Schwarz it suffices to show that

$\displaystyle \sum_{n \leq x} \frac{ | \frac{1}{\log \log P} \sum_{p \leq P: p|n} 1 - 1 |^2}{n} = o( \log x ).$

Expanding out the square, it suffices to show that

$\displaystyle \sum_{n \leq x} \frac{ (\frac{1}{\log \log P} \sum_{p \leq P: p|n} 1)^j}{n} = \log x + o( \log x )$

for ${j=0,1,2}$.

We just show the ${j=2}$ case, as the ${j=0,1}$ cases are similar (and easier). We rearrange the left-hand side as

$\displaystyle \frac{1}{(\log\log P)^2} \sum_{p_1, p_2 \leq P} \sum_{n \leq x: p_1,p_2|n} \frac{1}{n}.$

We can estimate the inner sum as ${(1+o(1)) \frac{1}{[p_1,p_2]} \log x}$. But a routine application of Mertens’ theorem (handling the diagonal case when ${p_1=p_2}$ separately) shows that

$\displaystyle \sum_{p_1, p_2 \leq P} \frac{1}{[p_1,p_2]} = (1+o(1)) (\log\log P)^2$

and the claim follows. $\Box$

Remark 4 As an alternative to the Turan-Kubilius inequality, one can use the Ramaré identity

$\displaystyle \sum_{p \leq P: p|n} \frac{1}{\# \{ p' \leq P: p'|n\} + 1} - 1 = 1_{(p,n)=1 \hbox{ for all } p \leq P}$

(see e.g. Section 17.3 of Friedlander-Iwaniec). This identity turns out to give superior quantitative results than the Turan-Kubilius inequality in applications; see the paper of Matomaki and Radziwiłł for an instance of this.

We now prove Theorem 2. Let ${Q}$ denote the left-hand side of (2); by the triangle inequality we have ${Q=O(1)}$. By Lemma 3 (for some ${P = x^{o(1)}}$ to be chosen later) and the triangle inequality we have

$\displaystyle \sum_{n \leq x} \frac{\frac{1}{\log \log P} \sum_{p \leq P: p|n} f(n)}{n} = Q \log x + o( \log x ).$

We rearrange the left-hand side as

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p} \sum_{m \leq x/p} \frac{f(m)}{m}.$

We now replace the constraint ${m \leq x/p}$ by ${m \leq x}$. The error incurred in doing so is

$\displaystyle O( \frac{1}{\log\log P} \sum_{p \leq P} \frac{1}{p} \sum_{x/P \leq m \leq x} \frac{1}{m} )$

which by Mertens’ theorem is ${O(\log P) = o( \log x )}$. Thus we have

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p} \sum_{m \leq x} \frac{f(m)}{m} = Q \log x + o( \log x ).$

But by definition of ${Q}$, we have ${\sum_{m \leq x} \frac{f(m)}{m} = Q \log x}$, thus

$\displaystyle [1 - \frac{1}{\log\log P} \sum_{p \leq P} \frac{f(p)}{p}] Q = o(1). \ \ \ \ \ (3)$

From Mertens’ theorem, the expression in brackets can be rewritten as

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{1 - f(p)}{p} + o(1)$

and so the real part of this expression is

$\displaystyle \frac{1}{\log\log P} \sum_{p \leq P} \frac{1 - \hbox{Re} f(p)}{p} + o(1).$

By (1), Mertens’ theorem and the hypothesis on ${f}$ we have

$\displaystyle \sum_{p \leq x^\varepsilon} \frac{(1 - \hbox{Re} f(p)) \log p}{p} \gg \log\log x^\varepsilon - O_\varepsilon(1)$

for any ${\varepsilon > 0}$. This implies that we can find ${P = x^{o(1)}}$ going to infinity such that

$\displaystyle \sum_{p \leq P} \frac{(1 - \hbox{Re} f(p)) \log p}{p} \gg (1-o(1))\log\log P$

and thus the expression in brackets has real part ${\gg 1-o(1)}$. The claim follows.

The Turan-Kubilius argument is certainly not the most efficient way to estimate sums such as ${\frac{1}{n} \sum_{n \leq x} f(n)}$. In the exercise below we give a significantly more accurate estimate that works when ${f}$ is non-negative.

Exercise 5 (Granville-Koukoulopoulos-Matomaki)

• (i) If ${g}$ is a completely multiplicative function with ${g(p) \in \{0,1\}}$ for all primes ${p}$, show that

$\displaystyle (e^{-\gamma}-o(1)) \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1} \leq \sum_{n \leq x} \frac{g(n)}{n} \leq \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1}.$

as ${x \rightarrow \infty}$. (Hint: for the upper bound, expand out the Euler product. For the lower bound, show that ${\sum_{n \leq x} \frac{g(n)}{n} \times \sum_{n \leq x} \frac{h(n)}{n} \ge \sum_{n \leq x} \frac{1}{n}}$, where ${h}$ is the completely multiplicative function with ${h(p) = 1-g(p)}$ for all primes ${p}$.)

• (ii) If ${g}$ is multiplicative and takes values in ${[0,1]}$, show that

$\displaystyle \sum_{n \leq x} \frac{g(n)}{n} \asymp \prod_{p \leq x} (1 - \frac{g(p)}{p})^{-1}$

$\displaystyle \asymp \exp( \sum_{p \leq x} \frac{g(p)}{p} )$

for all ${x \geq 1}$.

Now we turn to a very recent result of Matomaki and Radziwiłł on mean values of multiplicative functions in short intervals. For sake of illustration we specialise their results to the simpler case of the Liouville function ${\lambda}$, although their arguments actually work (with some additional effort) for arbitrary multiplicative functions of magnitude at most ${1}$ that are real-valued (or more generally, stay far from complex characters ${p \mapsto p^{it}}$). Furthermore, we give a qualitative form of their estimates rather than a quantitative one:

Theorem 6 (Matomaki-Radziwiłł, special case) Let ${X}$ be a parameter going to infinity, and let ${2 \leq h \leq X}$ be a quantity going to infinity as ${X \rightarrow \infty}$. Then for all but ${o(X)}$ of the integers ${x \in [X,2X]}$, one has

$\displaystyle \sum_{x \leq n \leq x+h} \lambda(n) = o( h ).$

Equivalently, one has

$\displaystyle \sum_{X \leq x \leq 2X} |\sum_{x \leq n \leq x+h} \lambda(n)|^2 = o( h^2 X ). \ \ \ \ \ (4)$

A simple sieving argument (see Exercise 18 of Supplement 4) shows that one can replace ${\lambda}$ by the Möbius function ${\mu}$ and obtain the same conclusion. See this recent note of Matomaki and Radziwiłł for a simple proof of their (quantitative) main theorem in this special case.

Of course, (4) improves upon the trivial bound of ${O( h^2 X )}$. Prior to this paper, such estimates were only known (using arguments similar to those in Section 3 of Notes 6) for ${h \geq X^{1/6+\varepsilon}}$ unconditionally, or for ${h \geq \log^A X}$ for some sufficiently large ${A}$ if one assumed the Riemann hypothesis. This theorem also represents some progress towards Chowla’s conjecture (discussed in Supplement 4) that

$\displaystyle \sum_{n \leq x} \lambda(n+h_1) \dots \lambda(n+h_k) = o( x )$

as ${x \rightarrow \infty}$ for any fixed distinct ${h_1,\dots,h_k}$; indeed, it implies that this conjecture holds if one performs a small amount of averaging in the ${h_1,\dots,h_k}$.

Below the fold, we give a “cheap” version of the Matomaki-Radziwiłł argument. More precisely, we establish

Theorem 7 (Cheap Matomaki-Radziwiłł) Let ${X}$ be a parameter going to infinity, and let ${1 \leq T \leq X}$. Then

$\displaystyle \int_X^{X^A} \left|\sum_{x \leq n \leq e^{1/T} x} \frac{\lambda(n)}{n}\right|^2\frac{dx}{x} = o\left( \frac{\log X}{T^2} \right), \ \ \ \ \ (5)$

for any fixed ${A>1}$.

Note that (5) improves upon the trivial bound of ${O( \frac{\log X}{T^2} )}$. Again, one can replace ${\lambda}$ with ${\mu}$ if desired. Due to the cheapness of Theorem 7, the proof will require few ingredients; the deepest input is the improved zero-free region for the Riemann zeta function due to Vinogradov and Korobov. Other than that, the main tools are the Turan-Kubilius result established above, and some Fourier (or complex) analysis.

In the previous set of notes, we saw how zero-density theorems for the Riemann zeta function, when combined with the zero-free region of Vinogradov and Korobov, could be used to obtain prime number theorems in short intervals. It turns out that a more sophisticated version of this type of argument also works to obtain prime number theorems in arithmetic progressions, in particular establishing the celebrated theorem of Linnik:

Theorem 1 (Linnik’s theorem) Let ${a\ (q)}$ be a primitive residue class. Then ${a\ (q)}$ contains a prime ${p}$ with ${p \ll q^{O(1)}}$.

In fact it is known that one can find a prime ${p}$ with ${p \ll q^{5}}$, a result of Xylouris. For sake of comparison, recall from Exercise 65 of Notes 2 that the Siegel-Walfisz theorem gives this theorem with a bound of ${p \ll \exp( q^{o(1)} )}$, and from Exercise 48 of Notes 2 one can obtain a bound of the form ${p \ll \phi(q)^2 \log^2 q}$ if one assumes the generalised Riemann hypothesis. The probabilistic random models from Supplement 4 suggest that one should in fact be able to take ${p \ll q^{1+o(1)}}$.

We will not aim to obtain the optimal exponents for Linnik’s theorem here, and follow the treatment in Chapter 18 of Iwaniec and Kowalski. We will in fact establish the following more quantitative result (a special case of a more powerful theorem of Gallagher), which splits into two cases, depending on whether there is an exceptional zero or not:

Theorem 2 (Quantitative Linnik theorem) Let ${a\ (q)}$ be a primitive residue class for some ${q \geq 2}$. For any ${x > 1}$, let ${\psi(x;q,a)}$ denote the quantity

$\displaystyle \psi(x;q,a) := \sum_{n \leq x: n=a\ (q)} \Lambda(n).$

Assume that ${x \geq q^C}$ for some sufficiently large ${C}$.

• (i) (No exceptional zero) If all the real zeroes ${\beta}$ of ${L}$-functions ${L(\cdot,\chi)}$ of real characters ${\chi}$ of modulus ${q}$ are such that ${1-\beta \gg \frac{1}{\log q}}$, then

$\displaystyle \psi(x;q,a) = \frac{x}{\phi(q)} ( 1 + O( \exp( - c \frac{\log x}{\log q} ) ) + O( \frac{\log^2 q}{q} ) )$

for all ${x \geq 1}$ and some absolute constant ${c>0}$.

• (ii) (Exceptional zero) If there is a zero ${\beta}$ of an ${L}$-function ${L(\cdot,\chi_1)}$ of a real character ${\chi_1}$ of modulus ${q}$ with ${\beta = 1 - \frac{\varepsilon}{\log q}}$ for some sufficiently small ${\varepsilon>0}$, then

$\displaystyle \psi(x;q,a) = \frac{x}{\phi(q)} ( 1 - \chi_1(a) \frac{x^{\beta-1}}{\beta} \ \ \ \ \ (1)$

$\displaystyle + O( \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) )$

$\displaystyle + O( \frac{\log^2 q}{q} ) )$

for all ${x \geq 1}$ and some absolute constant ${c>0}$.

The implied constants here are effective.

Note from the Landau-Page theorem (Exercise 54 from Notes 2) that at most one exceptional zero exists (if ${\varepsilon}$ is small enough). A key point here is that the error term ${O( \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) )}$ in the exceptional zero case is an improvement over the error term when no exceptional zero is present; this compensates for the potential reduction in the main term coming from the ${\chi_1(a) \frac{x^{\beta-1}}{\beta}}$ term. The splitting into cases depending on whether an exceptional zero exists or not turns out to be an essential technique in many advanced results in analytic number theory (though presumably such a splitting will one day become unnecessary, once the possibility of exceptional zeroes are finally eliminated for good).

Exercise 3 Assuming Theorem 2, and assuming ${x \geq q^C}$ for some sufficiently large absolute constant ${C}$, establish the lower bound

$\displaystyle \psi(x;a,q) \gg \frac{x}{\phi(q)}$

when there is no exceptional zero, and

$\displaystyle \psi(x;a,q) \gg \varepsilon \frac{x}{\phi(q)}$

when there is an exceptional zero ${\beta = 1 - \frac{\varepsilon}{\log q}}$. Conclude that Theorem 2 implies Theorem 1, regardless of whether an exceptional zero exists or not.

Remark 4 The Brun-Titchmarsh theorem (Exercise 33 from Notes 4), in the sharp form of Montgomery and Vaughan, gives that

$\displaystyle \pi(x; q, a) \leq 2 \frac{x}{\phi(q) \log (x/q)}$

for any primitive residue class ${a\ (q)}$ and any ${x \geq q}$. This is (barely) consistent with the estimate (1). Any lowering of the coefficient ${2}$ in the Brun-Titchmarsh inequality (with reasonable error terms), in the regime when ${x}$ is a large power of ${q}$, would then lead to at least some elimination of the exceptional zero case. However, this has not led to any progress on the Landau-Siegel zero problem (and may well be just a reformulation of that problem). (When ${x}$ is a relatively small power of ${q}$, some improvements to Brun-Titchmarsh are possible that are not in contradiction with the presence of an exceptional zero; see this paper of Maynard for more discussion.

Theorem 2 is deduced in turn from facts about the distribution of zeroes of ${L}$-functions. Recall from the truncated explicit formula (Exercise 45(iv) of Notes 2) with (say) ${T := q^2}$ that

$\displaystyle \sum_{n \leq x} \Lambda(n) \chi(n) = - \sum_{\hbox{Re}(\rho) > 3/4; |\hbox{Im}(\rho)| \leq q^2; L(\rho,\chi)=0} \frac{x^\rho}{\rho} + O( \frac{x}{q^2} \log^2 q)$

for any non-principal character ${\chi}$ of modulus ${q}$, where we assume ${x \geq q^C}$ for some large ${C}$; for the principal character one has the same formula with an additional term of ${x}$ on the right-hand side (as is easily deduced from Theorem 21 of Notes 2). Using the Fourier inversion formula

$\displaystyle 1_{n = a\ (q)} = \frac{1}{\phi(q)} \sum_{\chi\ (q)} \overline{\chi(a)} \chi(n)$

(see Theorem 69 of Notes 1), we thus have

$\displaystyle \psi(x;a,q) = \frac{x}{\phi(q)} ( 1 - \sum_{\chi\ (q)} \overline{\chi(a)} \sum_{\hbox{Re}(\rho) > 3/4; |\hbox{Im}(\rho)| \leq q^2; L(\rho,\chi)=0} \frac{x^{\rho-1}}{\rho}$

$\displaystyle + O( \frac{\log^2 q}{q} ) )$

and so it suffices by the triangle inequality (bounding ${1/\rho}$ very crudely by ${O(1)}$, as the contribution of the low-lying zeroes already turns out to be quite dominant) to show that

$\displaystyle \sum_{\chi\ (q)} \sum_{\sigma > 3/4; |t| \leq q^2; L(\sigma+it,\chi)=0} x^{\sigma-1} \ll \exp( - c \frac{\log x}{\log q} ) \ \ \ \ \ (2)$

when no exceptional zero is present, and

$\displaystyle \sum_{\chi\ (q)} \sum_{\sigma > 3/4; |t| \leq q^2; L(\sigma+it,\chi)=0; \sigma+it \neq \beta} x^{\sigma-1} \ll \exp( - c \frac{\log x}{\log q} \log \frac{1}{\varepsilon} ) \ \ \ \ \ (3)$

when an exceptional zero is present.

To handle the former case (2), one uses two facts about zeroes. The first is the classical zero-free region (Proposition 51 from Notes 2), which we reproduce in our context here:

Proposition 5 (Classical zero-free region) Let ${q, T \geq 2}$. Apart from a potential exceptional zero ${\beta}$, all zeroes ${\sigma+it}$ of ${L}$-functions ${L(\cdot,\chi)}$ with ${\chi}$ of modulus ${q}$ and ${|t| \leq T}$ are such that

$\displaystyle \sigma \leq 1 - \frac{c}{\log qT}$

for some absolute constant ${c>0}$.

Using this zero-free region, we have

$\displaystyle x^{\sigma-1} \ll \log x \int_{1/2}^{1-c/\log q} 1_{\alpha < \sigma} x^{\alpha-1}\ d\alpha$

whenever ${\sigma}$ contributes to the sum in (2), and so the left-hand side of (2) is bounded by

$\displaystyle \ll \log x \int_{1/2}^{1 - c/\log q} N( \alpha, q, q^2 ) x^{\alpha-1}\ d\alpha$

where we recall that ${N(\alpha,q,T)}$ is the number of zeroes ${\sigma+it}$ of any ${L}$-function of a character ${\chi}$ of modulus ${q}$ with ${\sigma \geq \alpha}$ and ${0 \leq t \leq T}$ (here we use conjugation symmetry to make ${t}$ non-negative, accepting a multiplicative factor of two).

In Exercise 25 of Notes 6, the grand density estimate

$\displaystyle N(\alpha,q,T) \ll (qT)^{4(1-\alpha)} \log^{O(1)}(qT) \ \ \ \ \ (4)$

is proven. If one inserts this bound into the above expression, one obtains a bound for (2) which is of the form

$\displaystyle \ll (\log^{O(1)} q) \exp( - c \frac{\log x}{\log q} ).$

Unfortunately this is off from what we need by a factor of ${\log^{O(1)} q}$ (and would lead to a weak form of Linnik’s theorem in which ${p}$ was bounded by ${O( \exp( \log^{O(1)} q ) )}$ rather than by ${q^{O(1)}}$). In the analogous problem for prime number theorems in short intervals, we could use the Vinogradov-Korobov zero-free region to compensate for this loss, but that region does not help here for the contribution of the low-lying zeroes with ${t = O(1)}$, which as mentioned before give the dominant contribution. Fortunately, it is possible to remove this logarithmic loss from the zero-density side of things:

Theorem 6 (Log-free grand density estimate) For any ${q, T > 1}$ and ${1/2 \leq \alpha \leq 1}$, one has

$\displaystyle N(\alpha,q,T) \ll (qT)^{O(1-\alpha)}.$

The implied constants are effective.

We prove this estimate below the fold. The proof follows the methods of the previous section, but one inserts various sieve weights to restrict sums over natural numbers to essentially become sums over “almost primes”, as this turns out to remove the logarithmic losses. (More generally, the trick of restricting to almost primes by inserting suitable sieve weights is quite useful for avoiding any unnecessary losses of logarithmic factors in analytic number theory estimates.)

Exercise 7 Use Theorem 6 to complete the proof of (2).

Now we turn to the case when there is an exceptional zero (3). The argument used to prove (2) applies here also, but does not gain the factor of ${\log \frac{1}{\varepsilon}}$ in the exponent. To achieve this, we need an additional tool, a version of the Deuring-Heilbronn repulsion phenomenon due to Linnik:

Theorem 8 (Deuring-Heilbronn repulsion phenomenon) Suppose ${q \geq 2}$ is such that there is an exceptional zero ${\beta = 1 - \frac{\varepsilon}{\log q}}$ with ${\varepsilon}$ small. Then all other zeroes ${\sigma+it}$ of ${L}$-functions of modulus ${q}$ are such that

$\displaystyle \sigma \leq 1 - c \frac{\log \frac{1}{\varepsilon}}{\log(q(2+|t|))}.$

In other words, the exceptional zero enlarges the classical zero-free region by a factor of ${\log \frac{1}{\varepsilon}}$. The implied constants are effective.

Exercise 9 Use Theorem 6 and Theorem 8 to complete the proof of (3), and thus Linnik’s theorem.

Exercise 10 Use Theorem 8 to give an alternate proof of (Tatuzawa’s version of) Siegel’s theorem (Theorem 62 of Notes 2). (Hint: if two characters have different moduli, then they can be made to have the same modulus by multiplying by suitable principal characters.)

Theorem 8 is proven by similar methods to that of Theorem 6, the basic idea being to insert a further weight of ${1 * \chi_1}$ (in addition to the sieve weights), the point being that the exceptional zero causes this weight to be quite small on the average. There is a strengthening of Theorem 8 due to Bombieri that is along the lines of Theorem 6, obtaining the improvement

$\displaystyle N'(\alpha,q,T) \ll \varepsilon (1 + \frac{\log T}{\log q}) (qT)^{O(1-\alpha)} \ \ \ \ \ (5)$

with effective implied constants for any ${1/2 \leq \alpha \leq 1}$ and ${T \geq 1}$ in the presence of an exceptional zero, where the prime in ${N'(\alpha,q,T)}$ means that the exceptional zero ${\beta}$ is omitted (thus ${N'(\alpha,q,T) = N(\alpha,q,T)-1}$ if ${\alpha \leq \beta}$). Note that the upper bound on ${N'(\alpha,q,T)}$ falls below one when ${\alpha > 1 - c \frac{\log \frac{1}{\varepsilon}}{\log(qT)}}$ for a sufficiently small ${c>0}$, thus recovering Theorem 8. Bombieri’s theorem can be established by the methods in this set of notes, and will be given as an exercise to the reader.

Remark 11 There are a number of alternate ways to derive the results in this set of notes, for instance using the Turan power sums method which is based on studying derivatives such as

$\displaystyle \frac{L'}{L}(s,\chi)^{(k)} = (-1)^k \sum_n \frac{\Lambda(n) \chi(n) \log^k n}{n^s}$

$\displaystyle \approx (-1)^{k+1} k! \sum_\rho \frac{1}{(s-\rho)^{k+1}}$

for ${\hbox{Re}(s)>1}$ and large ${k}$, and performing various sorts of averaging in ${k}$ to attenuate the contribution of many of the zeroes ${\rho}$. We will not develop this method here, but see for instance Chapter 9 of Montgomery’s book. See the text of Friedlander and Iwaniec for yet another approach based primarily on sieve-theoretic ideas.

Remark 12 When one optimises all the exponents, it turns out that the exponent in Linnik’s theorem is extremely good in the presence of an exceptional zero – indeed Friedlander and Iwaniec showed can even get a bound of the form ${p \ll q^{2-c}}$ for some ${c>0}$, which is even stronger than one can obtain from GRH! There are other places in which exceptional zeroes can be used to obtain results stronger than what one can obtain even on the Riemann hypothesis; for instance, Heath-Brown used the hypothesis of an infinite sequence of Siegel zeroes to obtain the twin prime conejcture.

In the previous set of notes, we studied upper bounds on sums such as ${|\sum_{N \leq n \leq N+M} n^{-it}|}$ for ${1 \leq M \leq N}$ that were valid for all ${t}$ in a given range, such as ${[T,2T]}$; this led in turn to upper bounds on the Riemann zeta ${\zeta(\sigma+it)}$ for ${t}$ in the same range, and for various choices of ${\sigma}$. While some improvement over the trivial bound of ${O(N)}$ was obtained by these methods, we did not get close to the conjectural bound of ${O( N^{1/2+o(1)})}$ that one expects from pseudorandomness heuristics (assuming that ${T}$ is not too large compared with ${N}$, e.g. ${T = O(N^{O(1)})}$.

However, it turns out that one can get much better bounds if one settles for estimating sums such as ${|\sum_{N \leq n \leq N+M} n^{-it}|}$, or more generally finite Dirichlet series (also known as Dirichlet polynomials) such as ${|\sum_n a_n n^{-it}|}$, for most values of ${t}$ in a given range such as ${[T,2T]}$. Equivalently, we will be able to get some control on the large values of such Dirichlet polynomials, in the sense that we can control the set of ${t}$ for which ${|\sum_n a_n n^{-it}|}$ exceeds a certain threshold, even if we cannot show that this set is empty. These large value theorems are often closely tied with estimates for mean values such as ${\frac{1}{T}\int_T^{2T} |\sum_n a_n n^{-it}|^{2k}\ dt}$ of a Dirichlet series; these latter estimates are thus known as mean value theorems for Dirichlet series. Our approach to these theorems will follow the same sort of methods used in Notes 3, in particular relying on the generalised Bessel inequality from those notes.

Our main application of the large value theorems for Dirichlet polynomials will be to control the number of zeroes of the Riemann zeta function ${\zeta(s)}$ (or the Dirichlet ${L}$-functions ${L(s,\chi)}$) in various rectangles of the form ${\{ \sigma+it: \sigma \geq \alpha, |t| \leq T \}}$ for various ${T > 1}$ and ${1/2 < \alpha < 1}$. These rectangles will be larger than the zero-free regions for which we can exclude zeroes completely, but we will often be able to limit the number of zeroes in such rectangles to be quite small. For instance, we will be able to show the following weak form of the Riemann hypothesis: as ${T \rightarrow \infty}$, a proportion ${1-o(1)}$ of zeroes of the Riemann zeta function in the critical strip with ${|\hbox{Im}(s)| \leq T}$ will have real part ${1/2+o(1)}$. Related to this, the number of zeroes with ${|\hbox{Im}(s)| \leq T}$ and ${|\hbox{Re}(s)| \geq \alpha}$ can be shown to be bounded by ${O( T^{O(1-\alpha)+o(1)} )}$ as ${T \rightarrow \infty}$ for any ${1/2 < \alpha < 1}$.

In the next set of notes we will use refined versions of these theorems to establish Linnik’s theorem on the least prime in an arithmetic progression.

Our presentation here is broadly based on Chapters 9 and 10 in Iwaniec and Kowalski, who give a number of more sophisticated large value theorems than the ones discussed here.