Today we develop a novel analytic framework linking the prime counting function π(x), the Prime Zeta function P(s), and the Riemann zeta function through Mellin-transform techniques and a Prime Counting Correction Formula.
Building from Abel summation, Stieltjes integration, and
distributional representations of prime counting, we derive integral identities
that express prime sums directly in terms of π(x) and establish analytic properties of the associated correction term.
These results culminate in a Mellin transform representation, revealing how
successive smoothing operators applied to the prime distribution encode
information about the zeros of the Riemann zeta function.
Lemma 1: Consider the integral
if 
then since 
, we have 
and
Thus,
and the two integrals
have the same convergence behavior.
Now
obviously converging if 
, that is, 
■
Lemma 2: 
counts the total number of primes less than or
equal to 
.
When 
is a prime number (
), 
increases by 
relative to 
, so:
When is composite number or , does not change, so:
The expression
where the summation is over prime numbers, is equal to
Instead of summing only over prime indices, we can sum over all
integers 
and use the indicator 
to filter out non-primes.
In “The Riemann Hypothesis Revealed” you find details about
Abel’s Identity:
where 
.
Take 
. Then 
. Thus, for an arithmetic function 
we have
Since 
■
Lemma 3: Another
expression of 
is:
where 
is the Heaviside step function: 
and 
. Since the distributional derivative
of the Heaviside function is
we get
where 
is the Dirac delta
function.
Note: The
formal definition of the Dirac delta function is often given by its behavior under
integration:
And for any function 
continuous at the origin:
This means that the Dirac delta function "picks out" the value of a function
at 
when integrated with
that function.
The derivative of a distribution is defined by 
which is consistent with
the formal definition of integration by parts and the behavior of the test
function at infinity.
So, for 
i.e., vanishing at
infinity,
■
Lemma 4:
Recall that for an ordinary integral 
, you add up 
weighted by tiny
intervals 
. For a Riemann- Stieltjes integral, 
, you add up 
weighted by the
increments 
.
Consider the Prime Zeta function
Take 
. Since is constant except at primes, its increments
are 
at each prime 
, and 
elsewhere. Therefore, the
Stieltjes integral becomes (see Lemma 3)
For 
,
Applying integration by parts for Riemann-Stieltjes integrals:
Thus,
because moving 
to the left gives 
.
Let 
and
The right-hand side integral
is the Mellin transform
of 
at 
( 
).
■
Lemma 5: Take the
right-hand side of
(see Lemma 4), and write
We cannot break this into two integrals like
unless these integrals converge. It turns out that this is the
case.
Two relatively long steps!
Step 1:
For 
Recall (see The Riemann Hypothesis Revealed), the Euler product
formula for the Riemann zeta function, valid for 
:
Since an absolutely convergent infinite product of non-zero
terms is strictly non-zero, 
for 
,
we can analyze its logarithm. We have 
. Thus, we can apply the Taylor
series expansion:
for 
, i.e.,
and
The theory of prime numbers is largely concerned with the
function 
, the number of primes not exceeding . We can express the outer sum over all integers
Indeed, (see Lemma 2 above)
the original expression
is a sum restricted purely to prime numbers . Using Lemma 2
Applying Abel’s Identity (Lemma 2) for and 
, we have
and
But 
for 
, so
Since 
Therefore,
and we have
implying
Step 2:
Let
is analytic and bounded for 
.
Indeed,
and for
hence
But for (see Lemma 1
above),
converges if 
.
So, for some 
and we have that the Prime
Counting Transform
is analytic for .
Thus, the Prime Counting Correction Formula
is valid.
Additionally,
■
Lemma 6: 
is analytic in the same region 
Write
Differentiating with respect to 
,
so
provided the differentiated integrand is dominated by an
integrable majorant.
Indeed, for
For
because we can bound
Since 
Thus,
But 
, hence
and by the Weirerstrass 
-test for improper integrals,
converges uniformly because we can bound its integrand by
an integrable function that is independent of 
.
Thus, differentiation under the
integral sign is justified, and is analytic in the same region .
■
A Mellin Transform Representation Derived from the Prime Counting Correction Formula
We can differentiate the Prime Counting Correction Formula
The result is:
or
Note: In Season 2 Episode 4, we saw that
where on the right-hand side, 
is the well-known arithmetic function called
the Mangoldt function, which returns 
when 
(i.e., a power of a prime
number), and zero otherwise.
Thus,
Integrating by parts (using 
where 
and 
)
Let
Then we write
Clearly 
. When infinity meets zero, the outcome usually depends entirely
on a "tug-of-war" (growth rate) between the two terms to see which
one wins. We have 
because 
is not just approaching
zero, it is identically equal to zero. Therefore, before 
ever has a chance to grow
infinitely large, it is multiplied directly by at every single point
near zero. That kills the product immediately, making the limit regardless of how large
or small is.
Additionally,
hence
because .
Thus,
Integrating by parts again
To verify that
observe that since 
for 
, 
. Let
Then 
for 
and 
regardless of . Additionally, 
as 
Thus,
So
Notice that
But
or
Replace with 
to get the Mellin
transform
and let for convenience
Therefore,
A very nice Mellin Transform.
In “The Riemann Hypothesis Revealed” I derive the Mellin
transform from Fourier Transform. The derivation shows that the Mellin
transform arises naturally when we shift perspective from additive to
multiplicative structures. By making the substitution 
, we convert multiplication into addition and reveal the Mellin
transform as a kind of “logarithmic Fourier transform”—one suited for analyzing
how functions behave under scaling rather than shifting.
The sequence of nested integrals
is built from which weighs the prime
distribution by its logarithmic density. Then and 
act as successive
smoothing or summation operators over the multiplicative scale.
Taking the Mellin transform of such nested structures naturally
yields meromorphic functions with poles encoding the zeros of Zeta
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