In this episode, we return to one of the most delicate—and
revealing—interfaces between complex analysis and analytic number theory: the
way zeros of the Riemann zeta function encode global information through
seemingly simple identities. Rather than treating these formulas as formal
manipulations, we slow down and examine why they work, which assumptions
are legitimate, and where symmetry plays a decisive role.
The discussion centers on the logarithmic derivative of the
completed zeta function and the constants that emerge when evaluating it at
special points. By carefully exploiting the functional equation and the pairing
of nontrivial zeros, we see how apparently asymmetric sums reorganize into
clean, meaningful expressions. What might look at first like an arbitrary
rearrangement turns out to be a structurally enforced consequence of analytic
continuation and absolute convergence.
Along the way, classical constants such as Euler’s constant
and familiar special functions reappear—not as isolated curiosities, but as
inevitable components of the analytic framework. The goal of this episode is
not merely to obtain a final formula, but to make transparent the logic that
leads there, highlighting how symmetry, convergence, and functional identities
cooperate behind the scenes.
This episode continues the broader theme of the series:
replacing compressed arguments with complete ones, and showing that many
“mysterious” results in analytic number theory become natural once every step
is laid out with care.
▪▪▪
Let’s start by recalling a few basic facts about logarithmic
derivatives. These are simple, powerful tools for understanding how zeros and
poles show up inside products like this:
What is the logarithmic derivative?
For a nonzero analytic function 
, the logarithmic derivative is
Even though 
itself is multi‑valued, its
derivative isn’t—different branches of the logarithm differ only by an additive
constant, and constants disappear when you differentiate. So 
is always a single‑valued analytic
function wherever 
has no zeros or poles.
Why is it useful?
Because logarithmic derivatives turn products into sums.
This is exactly what we want when dealing with infinite
products like the Hadamard product for 
.
To warm up, let’s explore the basic cases.
Logarithmic derivative of a product
Let
with each 
analytic and nonzero on
some domain.
Then
where each 
is a (locally)
single-valued branch and 
is a constant (coming
from branch choices). Differentiating:
So the logarithmic derivative of a finite product is the sum of the logarithmic derivatives of the factors. For infinite products, the same holds where the product converges and the sum of logarithmic derivatives converges normally. See The Riemann Hypothesis Revealed: Infinite products.
Single-valuedness of 
Even though 
is multi-valued (branches
differ by integer multiples of 
), any two branches differ by a constant:
Differentiating:
so
is independent of the branch and hence is a well-defined
single-valued analytic function wherever is analytic and nonzero.
Behavior at a zero of 
Suppose has a zero of multiplicity 
at 
. then locally
with 
analytic and 
.
Differentiate:
Then
Here 
is analytic near (since ). Thus:
has a simple pole at 
with residue .
Behavior at a pole of
Suppose has a pole of order at . the locally
with 
analytic and 
.
Differentiate:
Then
Again 
is analytic near , so
has a simple pole at
with residue 
.
Meromorphic case
If is meromorphic on a domain, then locally it is
analytic except at isolated poles. On a region avoiding zeros and poles, is analytic. At each zero or pole, we just saw
has at worst a simple pole. Therefore:
If 
is meromorphic,
then 
us meromorphic,
with only simple poles, located exactly at the zeros and poles of , with residues equal to the multiplicities (positive for
zeros, negative for poles).
______
After establishing that cleanly captures the
zero–pole structure of any meromorphic function—each zero or pole contributing
a simple pole to the logarithmic derivative with residue equal to its
multiplicity—we’re ready to apply this idea to a function of central importance
in analytic number theory.
The 
function: a cleaner way
to package 
A particularly convenient way to package the analytic properties
of the Riemann zeta function is through the auxiliary function 
, introduced and discussed in detail at the end of Analytic
Continuation of the Zeta Function in The Riemann Hypothesis Revealed.
It is defined by
By analytically continuing Zeta to the half-plane 
and using the symmetry 
, we concluded that is an entire function.
At this point, I assume the reader is familiar with the arguments
developed earlier in the book, especially the application of Hadamard’s
factorization theorem to 
Additionally you may
revisit the first two episodes of this season. Episode 2 and Episode 1
We also proved that 
has infinitely many zeros,
and admits the canonical product representation:
where 
denotes the non-trivial
zeros of the zeta function (see Existence of Zeta’s Non‑Trivial Zeros in
The Riemann Hypothesis Revealed).
The purpose of the present discussion is to move beyond these
foundational results and develop more advanced consequences that rely on the
material the reader has already studied.
Observe that the trivial zeros of 
do not correspond to
zeros of since they are canceled
by the poles of the gamma factor 
, as it is clear from the above
defining relation 
.
We begin with the definition
Taking the logarithmic derivative, we obtain
From our general principles:
is entire, and its zeros are exactly the nontrivial
zeros of 
. Therefore, 
is meromorphic with
only simple poles, located precisely at the nontrivial zeros 
, each with residue +1.
Now look at the right-hand side term:
has a simple pole at 
has a simple pole at 
has simple poles at the
poles of , i.e. at 
has a simple pole at with residue -1, simple
poles at every zero of (trivial and nontrivial),
each with residue +1.
Because is entire, all poles
coming from these pieces must cancel, except at the nontrivial zeros.
Concretely:
The poles at and the negative even
integers cancel between the 
-term, the term, and the trivial
zeros of .
The pole at cancels between and .
What’s left are simple poles at the nontrivial zeros , each with residue +1.
So we now have two complementary descriptions of 
:
Abstract:
meromorphic, simple poles at the nontrivial zeros , residue 1.
Concrete:
given by
Equating these two viewpoints—and
then integrating or testing against suitable functions—is exactly what leads to
the explicit formula we’re heading toward. That’s coming up in the next
episode, so don’t miss it!
Using the symmetry relation and differentiating
logarithmically, it follows that
In The Riemann Hypothesis Revealed, we also prove the
asymmetric form of the functional equation for the Riemann zeta function,
Taking the logarithmic derivative of this equation yields
Doing the same on the canonical representation of we write
and
where represents a non-trivial root of Zeta.
Using
and the logarithmic derivative of the identity 
which yields
we get
Thus,
By setting in 
and from
it follows
Since 
is real (in the book I
prove that 
),
However, as I state in the book if we assume a zero, the functional
equation gives that 
must also be a zero,
hence
because the map 
is just a reindexing of the same set and
the series 
is absolutely convergent,
so it’s safe to rearrange the index.
Thus,
Additionally,
may be rewritten as
Letting 
, and using the identities (taken from the book)
, where 
is the Euler constant,
and
we get exactly what we already know:
Epilogue
This episode shows how much structure
is hidden inside formulas that are often written down too quickly. By treating
convergence, symmetry, and functional identities with care, expressions
involving the zeros of the zeta function stop looking formal and start looking
inevitable. What emerges is a recurring theme of analytic number theory: once
the framework is set correctly, the constants and identities fall into place on
their own.
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