In this
episode, we return to the core ideas of entire functions and infinite products,
focusing on the subtle but crucial notions of order, genus, and canonical
products. Building directly on Hadamard’s theorem, we examine how the
growth of an entire function is encoded in the distribution of its zeros—and
how small changes in a product representation can fundamentally alter
convergence and analyticity. Along the way, we confront a typographical error
from the book, using it not as a distraction but as an opportunity to sharpen
intuition and clarify why the precise form of a Weierstrass product matters.
The goal of this episode is not only to apply the theory, but to understand
where it breaks if handled carelessly—and why that understanding is essential
in analytic number theory.
It will be
helpful to read the preceding episode first.
In The Riemann Hypothesis Revealed, we
considered an entire function 
having a zero of
multiplicity 
at 
and additional zeros 
arranged such that 
, where each distinct zero appears in the sequence exactly 
times if it has
multiplicity . We defined a sequence of integers 
satisfying
for all 
, and introduced the product
where
are called elementary factors.
Additionally, we established the
existence of an entire function 
such that
and with the assumption that all integers 
are equal to 
, for some fixed integer 
, we deduced that the product:
converges if 
. Finally, we defined the rank of
the product to be equal to 
, where is the smallest integer
for which and the equality 
holds.
If , in this representation of the entire function , is a polynomial of degree 
, we define a number 
called the genus of . Specifically, 
allowing 
.
The genus of an entire function is a measure of the growth rate of
its Weierstrass primary factors relative to its zeros. The order 
of an entire function is
a measure of how fast the function grows as 
.
The genus provides a more refined
description of the distribution of the zeros of a function. It is a key concept
in complex analysis for entire functions, reflecting the "minimal
complexity" needed to express the function via its zeros.
Usually, if for some integer 
, then is a candidate of the
genus of . The genus is the smallest integer for which the series 
converges, if of course 
. A low genus reflects the function’s moderate growth driven by
infinitely many zeros.
At the end of the section Hadamard’s
Theorem we concluded that
The genus
and the order
of an entire function satisfy
To explore
these ideas in greater depth, let us work through a sequence of exercises (taken
from the book) that clarify the role of order, genus, and canonical products.
1. Let
i.
Determine the order and genus of .
ii.
Verify that 
holds.
For 
zeros are at 
and 
. Since
for the genus must be 1.
Additionally
implies that 
. Recall
that
where 
. So from 
, we get .
Obviously
The
following, taken from the book, contains a typographical error—courtesy of the
ever-present demon of typography!
2. Let 
, and define:
i.
Show that is entire (I originally asked you to prove
that 
is entire, which is in fact false. As we shall
see, the correct function is 
)
ii.
Show that it has order and infinitely many zeros. (Of course, this
statement does not make sense if is not entire; this issue will be corrected in
what follows.)
To check
convergence, recall the standard criterion for an infinite product 
to converge uniformly on compact sets is (see
Season 2 episode 1)
Here,
Then
Since 
and , the series 
diverges.
So, the
product does not converge absolutely, but we can use the Weierstrass product
theorem.
For a sequence
of zeros 
a Weirstrass product is
Taking the
logarithm of the factor, we have
We will prove that the series
converges
absolutely for any fixed 
.
Recall that 
and therefore each series
is convergent.
The
Weierstrass product defines an entire function.
Indeed, for large
This comes
from setting 
So
Exponentiating:
Keeping only
the leading term:
Hence,
and
Let us now
examine the function 
in its incorrectly stated form.
Multiply and divide by 
and take the limit
This is the incorrectly stated function .
We cannot conclude
that the product
cancels the
factor
Indeed, for 
the sum 
diverges as 
and consequently
This decay
dominates and forces the entire expression to collapse to zero, rather than
yielding a cancellation.
So the
combined expression
does not converge
as
An independent
verification of the non‑convergence follows by taking logarithms, using the
previous computation of the product.
We have:
But
Then we expand
add 
to this:
sum over :
But as we saw,
the sum
exists and is
finite. The modified product
converges.
However, in the limit
the term 
diverges as (unless )
In light of
the above, the corrected product 
allows us to resolve the second part without
difficulty. The details are left as an exercise.
This episode
closes a chapter that is as much about method as it is about results. By
working carefully through order, genus, and canonical products—and by
confronting errors rather than glossing over them—we have seen how subtle
analytic issues reveal themselves only when every step is made explicit. This
is precisely where understanding deepens: not when everything works smoothly,
but when something fails and forces us to examine why.
One of the
central purposes of this blog is to provide ongoing support alongside
the book. The posts are not meant to replace the text, nor merely to repeat it,
but to stand next to it as a living companion. Here, corrections can be
addressed openly, alternative viewpoints explored, and intermediate steps
spelled out without compromise. When a typographical error appears, or when a
statement requires refinement, the blog becomes the place where the mathematics
is stabilized and clarified—carefully, rigorously, and without rushing the
reader forward.
Most
importantly, this support is continuous. Each episode builds on the
previous ones, revisiting ideas from different angles and reinforcing them
through concrete calculations and exercises. If something feels unclear at
first reading, that is not a failure—it is an invitation to engage more deeply,
knowing that further guidance is already waiting in the next post. Mathematics
is not mastered in a single pass, and this blog exists to accompany you through
that iterative process, step by step, until the structure finally clicks into
place.
Season 2 will
continue in this spirit: precise, patient, and responsive to the real
difficulties that arise when one learns complex analysis seriously. The goal is
not speed, but clarity—and the assurance that you are not working through these
ideas alone.
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