In the section Möbius Transformations of “The Riemann Hypothesis
Revealed” we claim that 
and 
are symmetric with
respect to the unit circle.
Recall that two points 
are said to be symmetric with respect to a circle or that 
is the inversion of 
with respect to the circle centered
at 
and radius 
, if both are on the same ray emanating from the center of the
circle and
In The Riemann Hypothesis Revealed, Möbius transformations are explored
through the lens of classical geometry, allowing the reader to develop a
genuine visual intuition for these remarkable mappings. Familiar geometric
ideas lead naturally to powerful results in complex analysis and analytic
number theory. For readers seeking an accessible yet rigorous path to deeper
mathematics, the book offers a rewarding and enjoyable experience.■
Let us now examine the inversion
more closely.
Lemma:
Inversion maps the half-plane
bijectively onto the disc
and vice versa.
Proof.
Let 
. Then
Since inversion is its own inverse,
it follows that the correspondence is one-to-one and onto. ∎
Since inversion maps a half-plane onto a disk, the following
lemma should come as no surprise.
Lemma: The Möbius transformation
maps the half plane 
bijectively onto the open
unit disk.
Moreover, since 
is a Möbius
transformation, it is conformal and therefore preserves angles between
intersecting curves.
Proof.
We have
Showing that maps the half-plane onto the unit disk. ∎
Schwarz’s Lemma: Let 
be the open unit disk in
the complex plane 
centered at the origin,
and let 
be holomorphic with
and 
on 
.
Then
and 
.
Moreover, if 
for some non-zero 
, or if 
, then
for some 
satisfying 
.
Proof.
A remarkably elegant proof, due to Constantin Carathéodory, consists of
a direct application of the Maximum Modulus Principle. The key idea is to
factor out the zero of 
at the origin and start
the resulting quotient.
Define
and extend 
to the origin by setting 
.
Since is holomorphic and , the singularity of 
at 
is removable.
Indeed, since is holomorphic and , differentiability at 
gives
Therefore
and hence
Because the limit exists and is finite, the singularity at is removable. Defining extends to a holomorphic function
on all .
Thus 
is holomorphic throughout
.
Fix 
. Since is holomorphic on a
neighborhood of the closed disc 
. the Maximum Modulus Principle implies that for every 
,
For 
because 
. Hence
Letting 
gives
Therefore,
which proves the first assertion. Evaluating at the origin
yields
For the extremal case, suppose that either for some nonzero , or . Then
or
Since on , the modulus of attains its maximum value
at an interior point of . By the Maximum Modulus Principle, must be constant:
with . Consequently,
which completes the proof. ■
Example:
For 
,
and . Schwarz’s lemma therefore applies and gives
which is obvious for
Moreover, 
if 
and 
for 
. Only attains equality, and
indeed 
is of the form 
with . ■
Example:
For real 
,
Hence Schwarz’s lemma applies and
This is precisely the extremal case. The lemma predicts that
every function with 
must be of this form. ■
Borel-Carathéodory Theorem: Let a
function be analytic on a closed disk of radius 
centered at the origin. Suppose that 
. Then, we have the
following inequality
where the norm on the left-hand side denotes the maximum value
of in the closed disc:
(the last equality is due to the Maximum Modulus Principle).
Proof.
Define
If is constant 
, the inequality follows from
Indeed, if 
then 
and
Since 
,
So we may assume is nonconstant.
Assume first that 
.
Since 
. we have 
. If 
then 
throughout the disk and
attains its maximum at an interior point. The Maximum Principle implies that 
, and therefore is constant, contrary to the assumption.
Recall from “The Riemann Hypothesis
Revealed: Poisson’s Integral formula” that if 
is analytic on and inside the circle 
, then by Cauchy's integral formula
we have:
Thus,
and for 
Thus, we may take 
and since with
for every 
. Because for , the image of the disk lies in the half-plane
By the previous lemma, the Möbius
transformation
maps this half-plane into the unit
disk. Moreover,
Therefore Schwarz's lemma yields
for all (define 
analytic on 
, satisfies 
and maps into . Schwarz’s lemma gives 
. Substituting 
yields 
).
Thus,
Take 
so
In the general case, we apply the above to 
giving
■
The theorem is named after Constantin
Carathéodory, one of Greece's most celebrated mathematicians. His
pioneering work in complex analysis helped establish many of the geometric
principles that now form part of the standard mathematical toolkit. Every time
we use a conformal map to transform a difficult region into a simpler one, we
are applying ideas that Carathéodory helped bring to maturity.
As shown in The
Riemann Hypothesis Revealed: Infinite Products, if is analytic function for
all finite values of 
and vanishes at the
points 
then for any finite
collection 
we may factor
where
is analytic and zero-free
in a neighborhood containing these zeros. Consequently,
expressing the logarithmic derivative as
the contribution from the zeros together with a regular analytic term.
More generally consider analytic in the closed
disk 
and take
where 
are the zeros of 
such that 
. The function
(where the
product runs over the zeros of satisfying 
, counted with multiplicity) is analytic for and not zero for 
.
Indeed, since each factor 
divides , the singularities are removable, so 
is analytic in .
The logarithmic derivative is
Also, since all zeros of in have been divided out,
any remaining zero 
of must satisfy 
and therefore cannot be a
zero of inside .
Now take
Clearly, for the circle 
, 
and
If 
In this case the inequality 
holds inside the circle
also, and hence the function
is analytic
for 
and 
.
Now assume that
for .
Then 
is analytic for , and
because
Borel-Carathéodory Theorem states that for
Thus,
for . Take for example 
, then
and
But (see The Derivatives of an Analytic Function)
so
or
for 
and are the zeros of such that 
.
So let’s recap:
Lemma: Let be analytic in the disk , and suppose that 
for 
throughout the disk. If ranges over the zeros of satisfying , then
for and an absolute constant .
Note: This lemma says that, deep inside the
disk, the logarithmic derivative 
is governed almost entirely by the nearby
zeros of .
The term
detects zeros through its simple poles, each zero contributing a term
The lemma shows that once these local contributions are removed,
what remains is a
comparatively small analytic function, bounded by 
.
Conceptually, this
means that in the interior of the disk the logarithmic derivative behaves like
the logarithmic derivative of a polynomial whose roots are precisely the zeros
of lying in . The influence of the rest of the
function is absorbed into a uniformly controlled error term.
Viewed another way, the lemma is a local factorization principle: after
accounting for the nearby zeros, resembles a zero-free analytic function whose
logarithmic derivative is small. This is precisely why estimates of 
are so often reduced to counting and locating
zeros (see A Small Taste
from My New Book, Season 2 Episode 7).
For applications to 
, the message is especially
important:
The logarithmic derivative is essentially a
weighted sum over nearby zeros, with all other effects contributing only a
controlled error.
That principle lies behind many formulas in analytic number theory.
Comments
Post a Comment