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Previous set of notes: Notes 3. Next set of notes: 246C Notes 1.
One of the great classical triumphs of complex analysis was in providing the first complete proof (by Hadamard and de la Vallée Poussin in 1896) of arguably the most important theorem in analytic number theory, the prime number theorem:
Theorem 1 (Prime number theorem) Letdenote the number of primes less than a given real number
. Then
(or in asymptotic notation,
as
).
(Actually, it turns out to be slightly more natural to replace the approximation in the prime number theorem by the logarithmic integral
, which happens to be a more precise approximation, but we will not stress this point here.)
The complex-analytic proof of this theorem hinges on the study of a key meromorphic function related to the prime numbers, the Riemann zeta function . Initially, it is only defined on the half-plane
:
Definition 2 (Riemann zeta function, preliminary definition) Letbe such that
. Then we define
Note that the series is locally uniformly convergent in the half-plane , so in particular
is holomorphic on this region. In previous notes we have already evaluated some special values of this function:
The Riemann zeta function has several remarkable properties, some of which we summarise here:
Theorem 3 (Basic properties of the Riemann zeta function)
- (i) (Euler product formula) For any
with
, we have
where the product is absolutely convergent (and locally uniform in
) and is over the prime numbers
.
- (ii) (Trivial zero-free region)
has no zeroes in the region
.
- (iii) (Meromorphic continuation)
has a unique meromorphic continuation to the complex plane (which by abuse of notation we also call
), with a simple pole at
and no other poles. Furthermore, the Riemann xi function
is an entire function of order
(after removing all singularities). The function
is an entire function of order one after removing the singularity at
.
- (iv) (Functional equation) After applying the meromorphic continuation from (iii), we have
for all
(excluding poles). Equivalently, we have
for all
. (The equivalence between the (5) and (6) is a routine consequence of the Euler reflection formula and the Legendre duplication formula, see Exercises 26 and 31 of Notes 1.)
Proof: We just prove (i) and (ii) for now, leaving (iii) and (iv) for later sections.
The claim (i) is an encoding of the fundamental theorem of arithmetic, which asserts that every natural number is uniquely representable as a product
over primes, where the
are natural numbers, all but finitely many of which are zero. Writing this representation as
, we see that
The claim (ii) is immediate from (i) since the Euler product is absolutely convergent and all terms are non-zero.
We remark that by sending to
in Theorem 3(i) we conclude that
The meromorphic continuation (iii) of the zeta function is initially surprising, but can be interpreted either as a manifestation of the extremely regular spacing of the natural numbers occurring in the sum (1), or as a consequence of various integral representations of
(or slight modifications thereof). We will focus in this set of notes on a particular representation of
as essentially the Mellin transform of the theta function
that briefly appeared in previous notes, and the functional equation (iv) can then be viewed as a consequence of the modularity of that theta function. This in turn was established using the Poisson summation formula, so one can view the functional equation as ultimately being a manifestation of Poisson summation. (For a direct proof of the functional equation via Poisson summation, see these notes.)
Henceforth we work with the meromorphic continuation of . The functional equation (iv), when combined with special values of
such as (2), gives some additional values of
outside of its initial domain
, most famously
From Theorem 3 and the non-vanishing nature of , we see that
has simple zeroes (known as trivial zeroes) at the negative even integers
, and all other zeroes (the non-trivial zeroes) inside the critical strip
. (The non-trivial zeroes are conjectured to all be simple, but this is hopelessly far from being proven at present.) As we shall see shortly, these latter zeroes turn out to be closely related to the distribution of the primes. The functional equation tells us that if
is a non-trivial zero then so is
; also, we have the identity
Conjecture 4 (Riemann hypothesis) All the non-trivial zeroes oflie on the critical line
.
This conjecture would have many implications in analytic number theory, particularly with regard to the distribution of the primes. Of course, it is far from proven at present, but the partial results we have towards this conjecture are still sufficient to establish results such as the prime number theorem.
Return now to the original region where . To take more advantage of the Euler product formula (3), we take complex logarithms to conclude that
The series and
that show up in the above formulae are examples of Dirichlet series, which are a convenient device to transform various sequences of arithmetic interest into holomorphic or meromorphic functions. Here are some more examples:
Exercise 5 (Standard Dirichlet series) Letbe a complex number with
.
- (i) Show that
.
- (ii) Show that
, where
is the divisor function of
(the number of divisors of
).
- (iii) Show that
, where
is the Möbius function, defined to equal
when
is the product of
distinct primes for some
, and
otherwise.
- (iv) Show that
, where
is the Liouville function, defined to equal
when
is the product of
(not necessarily distinct) primes for some
.
- (v) Show that
, where
is the holomorphic branch of the logarithm that is real for
, and with the convention that
vanishes for
.
- (vi) Use the fundamental theorem of arithmetic to show that the von Mangoldt function is the unique function
such that
for every positive integer
. Use this and (i) to provide an alternate proof of the identity (8). Thus we see that (8) is really just another encoding of the fundamental theorem of arithmetic.
Given the appearance of the von Mangoldt function , it is natural to reformulate the prime number theorem in terms of this function:
Theorem 6 (Prime number theorem, von Mangoldt form) One has(or in asymptotic notation,
as
).
Let us see how Theorem 6 implies Theorem 1. Firstly, for any , we can write
Exercise 7 Show that Theorem 1 conversely implies Theorem 6.
The alternate form (8) of the Euler product identity connects the primes (represented here via proxy by the von Mangoldt function) with the logarithmic derivative of the zeta function, and can be used as a starting point for describing further relationships between and the primes. Most famously, we shall see later in these notes that it leads to the remarkably precise Riemann-von Mangoldt explicit formula:
Theorem 8 (Riemann-von Mangoldt explicit formula) For any non-integer, we have
where
ranges over the non-trivial zeroes of
with imaginary part in
. Furthermore, the convergence of the limit is locally uniform in
.
Actually, it turns out that this formula is in some sense too precise; in applications it is often more convenient to work with smoothed variants of this formula in which the sum on the left-hand side is smoothed out, but the contribution of zeroes with large imaginary part is damped; see Exercise 22. Nevertheless, this formula clearly illustrates how the non-trivial zeroes of the zeta function influence the primes. Indeed, if one formally differentiates the above formula in
, one is led to the (quite nonrigorous) approximation
Comparing Theorem 8 with Theorem 6, it is natural to suspect that the key step in the proof of the latter is to establish the following slight but important extension of Theorem 3(ii), which can be viewed as a very small step towards the Riemann hypothesis:
Theorem 9 (Slight enlargement of zero-free region) There are no zeroes ofon the line
.
It is not quite immediate to see how Theorem 6 follows from Theorem 8 and Theorem 9, but we will demonstrate it below the fold.
Although Theorem 9 only seems like a slight improvement of Theorem 3(ii), proving it is surprisingly non-trivial. The basic idea is the following: if there was a zero at , then there would also be a different zero at
(note
cannot vanish due to the pole at
), and then the approximation (9) becomes
In fact, Theorem 9 is basically equivalent to the prime number theorem:
Exercise 10 For the purposes of this exercise, assume Theorem 6, but do not assume Theorem 9. For any non-zero real, show that
as
, where
denotes a quantity that goes to zero as
after being multiplied by
. Use this to derive Theorem 9.
This equivalence can help explain why the prime number theorem is remarkably non-trivial to prove, and why the Riemann zeta function has to be either explicitly or implicitly involved in the proof.
This post is only intended as the briefest of introduction to complex-analytic methods in analytic number theory; also, we have not chosen the shortest route to the prime number theorem, electing instead to travel in directions that particularly showcase the complex-analytic results introduced in this course. For some further discussion see this previous set of lecture notes, particularly Notes 2 and Supplement 3 (with much of the material in this post drawn from the latter).
Previous set of notes: Notes 2. Next set of notes: Notes 4.
On the real line, the quintessential examples of a periodic function are the (normalised) sine and cosine functions ,
, which are
-periodic in the sense that
What about periodic functions on the complex plane? We can start with singly periodic functions which obey a periodicity relationship
for all
in the domain and some period
; such functions can also be viewed as functions on the “additive cylinder”
(or equivalently
). We can rescale
as before. For holomorphic functions, we have the following characterisations:
Proposition 1 (Description of singly periodic holomorphic functions)In both cases, the coefficients
- (i) Every
-periodic entire function
has an absolutely convergent expansion
where
is the nome
, and the
are complex coefficients such that
Conversely, every doubly infinite sequence
of coefficients obeying (2) gives rise to a
-periodic entire function
via the formula (1).
- (ii) Every bounded
-periodic holomorphic function
on the upper half-plane
has an expansion
where the
are complex coefficients such that
Conversely, every infinite sequence
obeying (4) gives rise to a
-periodic holomorphic function
which is bounded away from the real axis (i.e., bounded on
for every
).
can be recovered from
by the Fourier inversion formula
for any
in
(in case (i)) or
(in case (ii)).
Proof: If is
-periodic, then it can be expressed as
for some function
on the “multiplicative cylinder”
, since the fibres of the map
are cosets of the integers
, on which
is constant by hypothesis. As the map
is a covering map from
to
, we see that
will be holomorphic if and only if
is. Thus
must have a Laurent series expansion
with coefficients
obeying (2), which gives (1), and the inversion formula (5) follows from the usual contour integration formula for Laurent series coefficients. The converse direction to (i) also follows by reversing the above arguments.
For part (ii), we observe that the map is also a covering map from
to the punctured disk
, so we can argue as before except that now
is a bounded holomorphic function on the punctured disk. By the Riemann singularity removal theorem (Exercise 35 of 246A Notes 3)
extends to be holomorphic on all of
, and thus has a Taylor expansion
for some coefficients
obeying (4). The argument now proceeds as with part (i).
The additive cylinder and the multiplicative cylinder
can both be identified (on the level of smooth manifolds, at least) with the geometric cylinder
, but we will not use this identification here.
Now let us turn attention to doubly periodic functions of a complex variable , that is to say functions
that obey two periodicity relations
Within the world of holomorphic functions, the collection of doubly periodic functions is boring:
Proposition 2 Letbe an entire doubly periodic function (with periods
linearly independent over
). Then
is constant.
In the language of Riemann surfaces, this proposition asserts that the torus is a non-hyperbolic Riemann surface; it cannot be holomorphically mapped non-trivially into a bounded subset of the complex plane.
Proof: The fundamental domain (up to boundary) enclosed by is compact, hence
is bounded on this domain, hence bounded on all of
by double periodicity. The claim now follows from Liouville’s theorem. (One could alternatively have argued here using the compactness of the torus
.
To obtain more interesting examples of doubly periodic functions, one must therefore turn to the world of meromorphic functions – or equivalently, holomorphic functions into the Riemann sphere . As it turns out, a particularly fundamental example of such a function is the Weierstrass elliptic function
Previous set of notes: Notes 1. Next set of notes: Notes 3.
In Exercise 5 (and Lemma 1) of 246A Notes 4 we already observed some links between complex analysis on the disk (or annulus) and Fourier series on the unit circle:
- (i) Functions
that are holomorphic on a disk
are expressed by a convergent Fourier series (and also Taylor series)
for
(so in particular
), where
conversely, every infinite sequenceof coefficients obeying (1) arises from such a function
.
- (ii) Functions
that are holomorphic on an annulus
are expressed by a convergent Fourier series (and also Laurent series)
, where
conversely, every doubly infinite sequenceof coefficients obeying (2) arises from such a function
.
- (iii) In the situation of (ii), there is a unique decomposition
where
extends holomorphically to
, and
extends holomorphically to
and goes to zero at infinity, and are given by the formulae
whereis any anticlockwise contour in
enclosing
, and and
whereis any anticlockwise contour in
enclosing
but not
.
This connection lets us interpret various facts about Fourier series through the lens of complex analysis, at least for some special classes of Fourier series. For instance, the Fourier inversion formula becomes the Cauchy-type formula for the Laurent or Taylor coefficients of
, in the event that the coefficients are doubly infinite and obey (2) for some
, or singly infinite and obey (1) for some
.
It turns out that there are similar links between complex analysis on a half-plane (or strip) and Fourier integrals on the real line, which we will explore in these notes.
We first fix a normalisation for the Fourier transform. If is an absolutely integrable function on the real line, we define its Fourier transform
by the formula
Exercise 1 (Fourier transform of Gaussian) Ifis a complex number with
and
is the Gaussian function
, show that the Fourier transform
is given by the Gaussian
, where we use the standard branch for
.
The Fourier transform has many remarkable properties. On the one hand, as long as the function is sufficiently “reasonable”, the Fourier transform enjoys a number of very useful identities, such as the Fourier inversion formula
Exercise 2 (Decay ofimplies regularity of
) Let
be an absolutely integrable function.
Hint: to establish holomorphicity in each of these cases, use Morera’s theorem and the Fubini-Tonelli theorem. For uniqueness, use analytic continuation, or (for part (iv)) the Schwartz reflection principle.
- (i) If
has super-exponential decay in the sense that
for all
and
(that is to say one has
for some finite quantity
depending only on
), then
extends uniquely to an entire function
. Furthermore, this function continues to be defined by (3).
- (ii) If
is supported on a compact interval
then the entire function
from (i) obeys the bounds
for
. In particular, if
is supported in
then
.
- (iii) If
obeys the bound
for all
and some
, then
extends uniquely to a holomorphic function
on the horizontal strip
, and obeys the bound
in this strip. Furthermore, this function continues to be defined by (3).
- (iv) If
is supported on
(resp.
), then there is a unique continuous extension of
to the lower half-plane
(resp. the upper half-plane
) which is holomorphic in the interior of this half-plane, and such that
uniformly as
(resp.
). Furthermore, this function continues to be defined by (3).
Later in these notes we will give a partial converse to part (ii) of this exercise, known as the Paley-Wiener theorem; there are also partial converses to the other parts of this exercise.
From (3) we observe the following intertwining property between multiplication by an exponential and complex translation: if is a complex number and
is an absolutely integrable function such that the modulated function
is also absolutely integrable, then we have the identity
The material in these notes is loosely adapted from Chapter 4 of Stein-Shakarchi’s “Complex Analysis”.
Previous set of notes: 246A Notes 5. Next set of notes: Notes 2.
— 1. Jensen’s formula —
Suppose is a non-zero rational function
, then by the fundamental theorem of algebra one can write
Exercise 1 Letbe a complex polynomial of degree
.
- (i) (Gauss-Lucas theorem) Show that the complex roots of
are contained in the closed convex hull of the complex roots of
.
- (ii) (Laguerre separation theorem) If all the complex roots of
are contained in a disk
, and
, then all the complex roots of
are also contained in
. (Hint: apply a suitable Möbius transformation to move
to infinity, and then apply part (i) to a polynomial that emerges after applying this transformation.)
There are a number of useful ways to extend these formulae to more general meromorphic functions than rational functions. Firstly there is a very handy “local” variant of (1) known as Jensen’s formula:
Theorem 2 (Jensen’s formula) Letbe a meromorphic function on an open neighbourhood of a disk
, with all removable singularities removed. Then, if
is neither a zero nor a pole of
, we have
where
and
range over the zeroes and poles of
respectively (counting multiplicity) in the disk
.
One can view (3) as a truncated (or localised) variant of (1). Note also that the summands are always non-positive.
Proof: By perturbing slightly if necessary, we may assume that none of the zeroes or poles of
(which form a discrete set) lie on the boundary circle
. By translating and rescaling, we may then normalise
and
, thus our task is now to show that
An important special case of Jensen’s formula arises when is holomorphic in a neighborhood of
, in which case there are no contributions from poles and one simply has
Exercise 3 Use (6) to give another proof of Liouville’s theorem: a bounded holomorphic functionon the entire complex plane is necessarily constant.
Exercise 4 Use Jensen’s formula to prove the fundamental theorem of algebra: a complex polynomialof degree
has exactly
complex zeroes (counting multiplicity), and can thus be factored as
for some complex numbers
with
. (Note that the fundamental theorem was invoked previously in this section, but only for motivational purposes, so the proof here is non-circular.)
Exercise 5 (Shifted Jensen’s formula) Letbe a meromorphic function on an open neighbourhood of a disk
, with all removable singularities removed. Show that
for all
in the open disk
that are not zeroes or poles of
, where
and
. (The function
appearing in the integrand is sometimes known as the Poisson kernel, particularly if one normalises so that
and
.)
Exercise 6 (Bounded type)
- (i) If
is a holomorphic function on
that is not identically zero, show that
.
- (ii) If
is a meromorphic function on
that is the ratio of two bounded holomorphic functions that are not identically zero, show that
. (Functions
of this form are said to be of bounded type and lie in the Nevanlinna class for the unit disk
.)
Exercise 7 (Smoothed out Jensen formula) Letbe a meromorphic function on an open set
, and let
be a smooth compactly supported function. Show that
where
range over the zeroes and poles of
(respectively) in the support of
. Informally argue why this identity is consistent with Jensen’s formula. (Note: as many of the functions involved here are not holomorphic, complex analysis tools are of limited use. Try using real variable tools such as Stokes theorem, Greens theorem, or integration by parts.)
When applied to entire functions , Jensen’s formula relates the order of growth of
near infinity with the density of zeroes of
. Here is a typical result:
Proposition 8 Letbe an entire function, not identically zero, that obeys a growth bound
for some
and all
. Then there exists a constant
such that
has at most
zeroes (counting multiplicity) for any
.
Entire functions that obey a growth bound of the form for every
and
(where
depends on
) are said to be of order at most
. The above theorem shows that for such functions that are not identically zero, the number of zeroes in a disk of radius
does not grow much faster than
. This is often a useful preliminary upper bound on the zeroes of entire functions, as the order of an entire function tends to be relatively easy to compute in practice.
Proof: First suppose that is non-zero. From (6) applied with
and
one has
Just as (3) and (7) give truncated variants of (1), we can create truncated versions of (2). The following crude truncation is adequate for many applications:
Theorem 9 (Truncated formula for log-derivative) Letbe a holomorphic function on an open neighbourhood of a disk
that is not identically zero on this disk. Suppose that one has a bound of the form
for some
and all
on the circle
. Let
be constants. Then one has the approximate formula
for all
in the disk
other than zeroes of
. Furthermore, the number of zeroes
in the above sum is
.
Proof: To abbreviate notation, we allow all implied constants in this proof to depend on .
We mimic the proof of Jensen’s formula. Firstly, we may translate and rescale so that and
, so we have
when
, and our main task is to show that
Suppose has a zero
with
. If we factor
, where
is the Blaschke product (5), then
Similarly, given a zero with
, we have
, so using Blaschke products to remove all of these zeroes also only affects the left-hand side of (8) by
(since the number of zeroes here is
), with
also modified by at most
. Thus we may assume in fact that
has no zeroes whatsoever within the unit disk. We may then also normalise
, then
for all
. By Jensen’s formula again, we have
Exercise 10
- (i) (Borel-Carathéodory theorem) If
is analytic on an open neighborhood of a disk
and
, show that
(Hint: one can normalise
,
,
, and
. Now
maps the unit disk to the half-plane
. Use a Möbius transformation to map the half-plane to the unit disk and then use the Schwarz lemma.)
- (ii) Use (i) to give an alternate way to conclude the proof of Theorem 9.
A variant of the above argument allows one to make precise the heuristic that holomorphic functions locally look like polynomials:
Exercise 11 (Local Weierstrass factorisation) Let the notation and hypotheses be as in Theorem 9. Then show thatfor all
in the disk
, where
is a polynomial whose zeroes are precisely the zeroes of
in
(counting multiplicity) and
is a holomorphic function on
of magnitude
and first derivative
on this disk. Furthermore, show that the degree of
is
.
Exercise 12 (Preliminary Beurling factorisation) Letdenote the space of bounded analytic functions
on the unit disk; this is a normed vector space with norm
- (i) If
is not identically zero, and
denote the zeroes of
in
counting multiplicity, show that
and
- (ii) Let the notation be as in (i). If we define the Blaschke product
where
is the order of vanishing of
at zero, show that this product converges absolutely to a holomorphic function on
, and that
for all
. (It may be easier to work with finite Blaschke products first to obtain this bound.)
- (iii) Continuing the notation from (i), establish a factorisation
for some holomorphic function
with
for all
.
- (iv) (Theorem of F. and M. Riesz, special case) If
extends continuously to the boundary
, show that the set
has zero measure.
Remark 13 The factorisation (iii) can be refined further, withbeing the Poisson integral of some finite measure on the unit circle. Using the Lebesgue decomposition of this finite measure into absolutely continuous parts one ends up factorising
functions into “outer functions” and “inner functions”, giving the Beurling factorisation of
. There are also extensions to larger spaces
than
(which are to
as
is to
), known as Hardy spaces. We will not discuss this topic further here, but see for instance this text of Garnett for a treatment.
Exercise 14 (Littlewood’s lemma) Letbe holomorphic on an open neighbourhood of a rectangle
for some
and
, with
non-vanishing on the boundary of the rectangle. Show that
where
ranges over the zeroes of
inside
(counting multiplicity) and one uses a branch of
which is continuous on the upper, lower, and right edges of
. (This lemma is a popular tool to explore the zeroes of Dirichlet series such as the Riemann zeta function.)
Just a short announcement that next quarter I will be continuing the recently concluded 246A complex analysis class as 246B. Topics I plan to cover:
- Schwartz-Christoffel transformations and the uniformisation theorem (using the remainder of the 246A notes);
- Jensen’s formula and factorisation theorems (particularly Weierstrass and Hadamard); the Gamma function;
- Connections with the Fourier transform on the real line;
- Elliptic functions and their relatives;
- (if time permits) the Riemann zeta function and the prime number theorem.
Notes for the later material will appear on this blog in due course.
Consider a disk in the complex plane. If one applies an affine-linear map
to this disk, one obtains
Theorem 1 (Holomorphic images of disks) Letbe a disk in the complex plane, and
be a holomorphic function with
.
- (i) (Open mapping theorem or inverse function theorem)
contains a disk
for some
. (In fact there is even a holomorphic right inverse of
from
to
.)
- (ii) (Bloch theorem)
contains a disk
for some absolute constant
and some
. (In fact there is even a holomorphic right inverse of
from
to
.)
- (iii) (Koebe quarter theorem) If
is injective, then
contains the disk
.
- (iv) If
is a polynomial of degree
, then
contains the disk
.
- (v) If one has a bound of the form
for all
and some
, then
contains the disk
for some absolute constant
. (In fact there is holomorphic right inverse of
from
to
.)
Parts (i), (ii), (iii) of this theorem are standard, as indicated by the given links. I found part (iv) as (a consequence of) Theorem 2 of this paper of Degot, who remarks that it “seems not already known in spite of its simplicity”; an equivalent form of this result also appears in Lemma 4 of this paper of Miller. The proof is simple:
Proof: (Proof of (iv)) Let , then we have a lower bound for the log-derivative of
at
:
The constant in (iv) is completely sharp: if
and
is non-zero then
contains the disk
Part (v) is implicit in the standard proof of Bloch’s theorem (part (ii)), and is easy to establish:
Proof: (Proof of (v)) From the Cauchy inequalities one has for
, hence by Taylor’s theorem with remainder
for
. By Rouche’s theorem, this implies that the function
has a unique zero in
for any
, if
is a sufficiently small absolute constant. The claim follows.
Note that part (v) implies part (i). A standard point picking argument also lets one deduce part (ii) from part (v):
Proof: (Proof of (ii)) By shrinking slightly if necessary we may assume that
extends analytically to the closure of the disk
. Let
be the constant in (v) with
; we will prove (iii) with
replaced by
. If we have
for all
then we are done by (v), so we may assume without loss of generality that there is
such that
. If
for all
then by (v) we have
Here is another classical result stated by Alexander (and then proven by Kakeya and by Szego, but also implied to a classical theorem of Grace and Heawood) that is broadly compatible with parts (iii), (iv) of the above theorem:
Proposition 2 Letbe a disk in the complex plane, and
be a polynomial of degree
with
for all
. Then
is injective on
.
The radius is best possible, for the polynomial
has
non-vanishing on
, but one has
, and
lie on the boundary of
.
If one narrows slightly to
then one can quickly prove this proposition as follows. Suppose for contradiction that there exist distinct
with
, thus if we let
be the line segment contour from
to
then
. However, by assumption we may factor
where all the
lie outside of
. Elementary trigonometry then tells us that the argument of
only varies by less than
as
traverses
, hence the argument of
only varies by less than
. Thus
takes values in an open half-plane avoiding the origin and so it is not possible for
to vanish.
To recover the best constant of requires some effort. By taking contrapositives and applying an affine rescaling and some trigonometry, the proposition can be deduced from the following result, known variously as the Grace-Heawood theorem or the complex Rolle theorem.
Proposition 3 (Grace-Heawood theorem) Letbe a polynomial of degree
such that
. Then
contains a zero in the closure of
.
This is in turn implied by a remarkable and powerful theorem of Grace (which we shall prove shortly). Given two polynomials of degree at most
, define the apolar form
by
Theorem 4 (Grace’s theorem) Letbe a circle or line in
, dividing
into two open connected regions
. Let
be two polynomials of degree at most
, with all the zeroes of
lying in
and all the zeroes of
lying in
. Then
.
(Contrapositively: if , then the zeroes of
cannot be separated from the zeroes of
by a circle or line.)
Indeed, a brief calculation reveals the identity
The same method of proof gives the following nice consequence:
Theorem 5 (Perpendicular bisector theorem) Letbe a polynomial such that
for some distinct
. Then the zeroes of
cannot all lie on one side of the perpendicular bisector of
. For instance, if
, then the zeroes of
cannot all lie in the halfplane
or the halfplane
.
I’d be interested in seeing a proof of this latter theorem that did not proceed via Grace’s theorem.
Now we give a proof of Grace’s theorem. The case can be established by direct computation, so suppose inductively that
and that the claim has already been established for
. Given the involvement of circles and lines it is natural to suspect that a Möbius transformation symmetry is involved. This is indeed the case and can be made precise as follows. Let
denote the vector space of polynomials
of degree at most
, then the apolar form is a bilinear form
. Each translation
on the complex plane induces a corresponding map on
, mapping each polynomial
to its shift
. We claim that the apolar form is invariant with respect to these translations:
Next, we see that the inversion map also induces a corresponding map on
, mapping each polynomial
to its inversion
. From (1) we see that this map also (projectively) preserves the apolar form:
Starting on Oct 2, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence at the math department here at UCLA. This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular focusing on the basic properties of holomorphic functions such as the Cauchy and residue theorems, the classification of singularities, and the maximum principle, but there will be more of an emphasis on rigour, generalisation and abstraction, and connections with other parts of mathematics. The main text I will be using for this course is Stein-Shakarchi (with Ahlfors as a secondary text), but I will also be using the blog lecture notes I wrote the last time I taught this course in 2016. At this time I do not expect to significantly deviate from my past lecture notes, though I do not know at present how different the pace will be this quarter when the course is taught remotely. As with my 247B course last spring, the lectures will be open to the public, though other coursework components will be restricted to enrolled students.
This set of notes discusses aspects of one of the oldest questions in Fourier analysis, namely the nature of convergence of Fourier series.
If is an absolutely integrable function, its Fourier coefficients
are defined by the formula
What if is not smooth, but merely lies in an
class for some
? The Fourier coefficients
remain well-defined, as do the partial summation operators
. The question of convergence in norm is relatively easy to settle:
Exercise 1
- (i) If
and
, show that
converges in
norm to
. (Hint: first use the boundedness of the Hilbert transform to show that
is bounded in
uniformly in
.)
- (ii) If
or
, show that there exists
such that the sequence
is unbounded in
(so in particular it certainly does not converge in
norm to
. (Hint: first show that
is not bounded in
uniformly in
, then apply the uniform boundedness principle in the contrapositive.)
The question of pointwise almost everywhere convergence turned out to be a significantly harder problem:
Theorem 2 (Pointwise almost everywhere convergence)
Note from Hölder’s inequality that contains
for all
, so Carleson’s theorem covers the
case of Hunt’s theorem. We remark that the precise threshold near
between Kolmogorov-type divergence results and Carleson-Hunt pointwise convergence results, in the category of Orlicz spaces, is still an active area of research; see this paper of Lie for further discussion.
Carleson’s theorem in particular was a surprisingly difficult result, lying just out of reach of classical methods (as we shall see later, the result is much easier if we smooth either the function or the summation method
by a tiny bit). Nowadays we realise that the reason for this is that Carleson’s theorem essentially contains a frequency modulation symmetry in addition to the more familiar translation symmetry and dilation symmetry. This basically rules out the possibility of attacking Carleson’s theorem with tools such as Calderón-Zygmund theory or Littlewood-Paley theory, which respect the latter two symmetries but not the former. Instead, tools from “time-frequency analysis” that essentially respect all three symmetries should be employed. We will illustrate this by giving a relatively short proof of Carleson’s theorem due to Lacey and Thiele. (There are other proofs of Carleson’s theorem, including Carleson’s original proof, its modification by Hunt, and a later time-frequency proof by Fefferman; see Remark 18 below.)
In contrast to previous notes, in this set of notes we shall focus exclusively on Fourier analysis in the one-dimensional setting for simplicity of notation, although all of the results here have natural extensions to higher dimensions. Depending on the physical context, one can view the physical domain
as representing either space or time; we will mostly think in terms of the former interpretation, even though the standard terminology of “time-frequency analysis”, which we will make more prominent use of in later notes, clearly originates from the latter.
In previous notes we have often performed various localisations in either physical space or Fourier space , for instance in order to take advantage of the uncertainty principle. One can formalise these operations in terms of the functional calculus of two basic operations on Schwartz functions
, the position operator
defined by
and the momentum operator , defined by
(The terminology comes from quantum mechanics, where it is customary to also insert a small constant on the right-hand side of (1) in accordance with de Broglie’s law. Such a normalisation is also used in several branches of mathematics, most notably semiclassical analysis and microlocal analysis, where it becomes profitable to consider the semiclassical limit
, but we will not emphasise this perspective here.) The momentum operator can be viewed as the counterpart to the position operator, but in frequency space instead of physical space, since we have the standard identity
for any and
. We observe that both operators
are formally self-adjoint in the sense that
for all , where we use the
Hermitian inner product
Clearly, for any polynomial of one real variable
(with complex coefficients), the operator
is given by the spatial multiplier operator
and similarly the operator is given by the Fourier multiplier operator
Inspired by this, if is any smooth function that obeys the derivative bounds
for all and
(that is to say, all derivatives of
grow at most polynomially), then we can define the spatial multiplier operator
by the formula
one can easily verify from several applications of the Leibniz rule that maps Schwartz functions to Schwartz functions. We refer to
as the symbol of this spatial multiplier operator. In a similar fashion, we define the Fourier multiplier operator
associated to the symbol
by the formula
For instance, any constant coefficient linear differential operators can be written in this notation as
however there are many Fourier multiplier operators that are not of this form, such as fractional derivative operators for non-integer values of
, which is a Fourier multiplier operator with symbol
. It is also very common to use spatial cutoffs
and Fourier cutoffs
for various bump functions
to localise functions in either space or frequency; we have seen several examples of such cutoffs in action in previous notes (often in the higher dimensional setting
).
We observe that the maps and
are ring homomorphisms, thus for instance
and
for any obeying the derivative bounds (2); also
is formally adjoint to
in the sense that
for , and similarly for
and
. One can interpret these facts as part of the functional calculus of the operators
, which can be interpreted as densely defined self-adjoint operators on
. However, in this set of notes we will not develop the spectral theory necessary in order to fully set out this functional calculus rigorously.
In the field of PDE and ODE, it is also very common to study variable coefficient linear differential operators
where the are now functions of the spatial variable
obeying the derivative bounds (2). A simple example is the quantum harmonic oscillator Hamiltonian
. One can rewrite this operator in our notation as
and so it is natural to interpret this operator as a combination of both the position operator
and the momentum operator
, where the symbol
this operator is the function
Indeed, from the Fourier inversion formula
for any we have
and hence on multiplying by and summing we have
Inspired by this, we can introduce the Kohn-Nirenberg quantisation by defining the operator by the formula
whenever and
is any smooth function obeying the derivative bounds
for all and
(note carefully that the exponent in
on the right-hand side is required to be uniform in
). This quantisation clearly generalises both the spatial multiplier operators
and the Fourier multiplier operators
defined earlier, which correspond to the cases when the symbol
is a function of
only or
only respectively. Thus we have combined the physical space
and the frequency space
into a single domain, known as phase space
. The term “time-frequency analysis” encompasses analysis based on decompositions and other manipulations of phase space, in much the same way that “Fourier analysis” encompasses analysis based on decompositions and other manipulations of frequency space. We remark that the Kohn-Nirenberg quantization is not the only choice of quantization one could use; see Remark 19 below.
In principle, the quantisations are potentially very useful for such tasks as inverting variable coefficient linear operators, or to localize a function simultaneously in physical and Fourier space. However, a fundamental difficulty arises: map from symbols
to operators
is now no longer a ring homomorphism, in particular
in general. Fundamentally, this is due to the fact that pointwise multiplication of symbols is a commutative operation, whereas the composition of operators such as and
does not necessarily commute. This lack of commutativity can be measured by introducing the commutator
of two operators , and noting from the product rule that
(In the language of Lie groups and Lie algebras, this tells us that are (up to complex constants) the standard Lie algebra generators of the Heisenberg group.) From a quantum mechanical perspective, this lack of commutativity is the root cause of the uncertainty principle that prevents one from simultaneously localizing in both position and momentum past a certain point. Here is one basic way of formalising this principle:
Exercise 2 (Heisenberg uncertainty principle) For any
and
, show that
(Hint: evaluate the expression
in two different ways and apply the Cauchy-Schwarz inequality.) Informally, this exercise asserts that the spatial uncertainty
and the frequency uncertainty
of a function obey the Heisenberg uncertainty relation
.
Nevertheless, one still has the correspondence principle, which asserts that in certain regimes (which, with our choice of normalisations, corresponds to the high-frequency regime), quantum mechanics continues to behave like a commutative theory, and one can sometimes proceed as if the operators (and the various operators
constructed from them) commute up to “lower order” errors. This can be formalised using the pseudodifferential calculus, which we give below the fold, in which we restrict the symbol
to certain “symbol classes” of various orders (which then restricts
to be pseudodifferential operators of various orders), and obtains approximate identities such as
where the error between the left and right-hand sides is of “lower order” and can in fact enjoys a useful asymptotic expansion. As a first approximation to this calculus, one can think of functions as having some sort of “phase space portrait”
which somehow combines the physical space representation
with its Fourier representation
, and pseudodifferential operators
behave approximately like “phase space multiplier operators” in this representation in the sense that
Unfortunately the uncertainty principle (or the non-commutativity of and
) prevents us from making these approximations perfectly precise, and it is not always clear how to even define a phase space portrait
of a function
precisely (although there are certain popular candidates for such a portrait, such as the FBI transform (also known as the Gabor transform in signal processing literature), or the Wigner quasiprobability distribution, each of which have some advantages and disadvantages). Nevertheless even if the concept of a phase space portrait is somewhat fuzzy, it is of great conceptual benefit both within mathematics and outside of it. For instance, the musical score one assigns a piece of music can be viewed as a phase space portrait of the sound waves generated by that music.
To complement the pseudodifferential calculus we have the basic Calderón-Vaillancourt theorem, which asserts that pseudodifferential operators of order zero are Calderón-Zygmund operators and thus bounded on for
. The standard proof of this theorem is a classic application of one of the basic techniques in harmonic analysis, namely the exploitation of almost orthogonality; the proof we will give here will achieve this through the elegant device of the Cotlar-Stein lemma.
Pseudodifferential operators (especially when generalised to higher dimensions ) are a fundamental tool in the theory of linear PDE, as well as related fields such as semiclassical analysis, microlocal analysis, and geometric quantisation. There is an even wider class of operators that is also of interest, namely the Fourier integral operators, which roughly speaking not only approximately multiply the phase space portrait
of a function by some multiplier
, but also move the portrait around by a canonical transformation. However, the development of theory of these operators is beyond the scope of these notes; see for instance the texts of Hormander or Eskin.
This set of notes is only the briefest introduction to the theory of pseudodifferential operators. Many texts are available that cover the theory in more detail, for instance this text of Taylor.
The square root cancellation heuristic, briefly mentioned in the preceding set of notes, predicts that if a collection of complex numbers have phases that are sufficiently “independent” of each other, then
similarly, if are a collection of functions in a Lebesgue space
that oscillate “independently” of each other, then we expect
We have already seen one instance in which this heuristic can be made precise, namely when the phases of are randomised by a random sign, so that Khintchine’s inequality (Lemma 4 from Notes 1) can be applied. There are other contexts in which a square function estimate
or a reverse square function estimate
(or both) are known or conjectured to hold. For instance, the useful Littlewood-Paley inequality implies (among other things) that for any , we have the reverse square function estimate
whenever the Fourier transforms of the
are supported on disjoint annuli
, and we also have the matching square function estimate
if there is some separation between the annuli (for instance if the are
-separated). We recall the proofs of these facts below the fold. In the
case, we of course have Pythagoras’ theorem, which tells us that if the
are all orthogonal elements of
, then
In particular, this identity holds if the have disjoint Fourier supports in the sense that their Fourier transforms
are supported on disjoint sets. For
, the technique of bi-orthogonality can also give square function and reverse square function estimates in some cases, as we shall also see below the fold.
In recent years, it has begun to be realised that in the regime , a variant of reverse square function estimates such as (1) is also useful, namely decoupling estimates such as
(actually in practice we often permit small losses such as on the right-hand side). An estimate such as (2) is weaker than (1) when
(or equal when
), as can be seen by starting with the triangle inequality
and taking the square root of both side to conclude that
However, the flip side of this weakness is that (2) can be easier to prove. One key reason for this is the ability to iterate decoupling estimates such as (2), in a way that does not seem to be possible with reverse square function estimates such as (1). For instance, suppose that one has a decoupling inequality such as (2), and furthermore each can be split further into components
for which one has the decoupling inequalities
Then by inserting these bounds back into (2) we see that we have the combined decoupling inequality
This iterative feature of decoupling inequalities means that such inequalities work well with the method of induction on scales, that we introduced in the previous set of notes.
In fact, decoupling estimates share many features in common with restriction theorems; in addition to induction on scales, there are several other techniques that first emerged in the restriction theory literature, such as wave packet decompositions, rescaling, and bilinear or multilinear reductions, that turned out to also be well suited to proving decoupling estimates. As with restriction, the curvature or transversality of the different Fourier supports of the will be crucial in obtaining non-trivial estimates.
Strikingly, in many important model cases, the optimal decoupling inequalities (except possibly for epsilon losses in the exponents) are now known. These estimates have in turn had a number of important applications, such as establishing certain discrete analogues of the restriction conjecture, or the first proof of the main conjecture for Vinogradov mean value theorems in analytic number theory.
These notes only serve as a brief introduction to decoupling. A systematic exploration of this topic can be found in this recent text of Demeter.
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