Today we
redraw our attention to Perron’s formula, a powerful bridge between Dirichlet
series and arithmetic sums.
Starting from
a general Dirichlet series, we develop the machinery needed to recover partial
sums via complex integration.
We then
present a fully rigorous proof—one you won’t want to miss.
Let 
and
be a Dirichlet
series. There exists a real number 
(the abscissa of convergence) such that the
series converges for all 
and diverges for all 
. On the open half-plane the function 
is analytic. In The Riemann Hypothesis Revealed, the treatment of Dirichlet series is intentionally concise.
Rather than pursuing a general theory, I focus on convergence established
through bounded partial sums,
for some 
. Under this condition,
one obtains a proof of the following theorem (see The Riemann Hypothesis Revealed: Dirichlet Series).
A Dirichlet series, 
with bounded
partial sums for some 
,
has a half-plane of convergence consisting of all complex
numbers such that 
, where 
is the real part
of 
and 
is the abscissa of
convergence. Within any closed and bounded subset (i.e., a compact subset) of
this half-plane of convergence the series converges uniformly.
Let’s start
with the following exercise (from the book):
Exercise: Assuming the
convergence of 
with 
, prove that
where 
. Hint: Take Abel’s
summation formula with 
. From this deduce that 
Solution: Let 
. Abel’s summation formula (see The Riemann Hypothesis Revealed)
states that
Let 
, 
. Then since 
We are given that the Dirichlet series converges. Hence the “tail” term must vanish:
Applying this
limit to our equation:
Thus, we
arrive at the desired identity:
For 
notice that 
and for 
(
has 
) and the comparison test, the
integral
converges (
).
Note that the
condition implies that 
is bounded, which in turn guarantees the
existence of the integral
Indeed, if we may take 
. Convergence then ensures that the partial sums
are bounded, so there exists 
such that
A classical
identity in complex analysis expresses the Heaviside step function as the inverse Mellin transform of 
. Specifically, the integral
i.e., equals 
for 
, 
for 
, and 
at 
This formula (see The Riemann Hypothesis Revealed for a proof), plays a central role in explicit formulas in analytic
number theory, where constant and slowly varying analytic terms yield
step-function contributions in the variable 
.
Recalling The Riemann Hypothesis Revealed (or any standard reference), the Mellin transform and its inverse
are given by
Thus the inverse Mellin transform satisfies 
.
The key is
understanding that this integral is an inverse Mellin Transform.
For this let’s define the
slightly modified Heaviside step function shifted to the right by 
:
Then the Mellin transform of 
is:
This integral converges for 
and gives:
The inverse Mellin transform
for 
is:
Now let 
and note that for 
and the formula to be proved is
Τhis is an inverse Mellin
transform.
You may find the proof in the book.
Note: The
preceding paragraph contained an incorrect argument in the original text—an
all-too-human error made during late-night writing. This correction (shown here
in blue) will be included in the next updated edition on Amazon.
The integral
plays a key
role in the derivation of the well-known Perron’s formula
The prime on the summation indicates
that the last term of the sum must be multiplied by when is an integer.
Additionally
is absolutely convergent for 
and 
As I state in the book, from
we get:
If we substitute
into the integral
This argument
is intentionally motivational rather than complete. Its role is not to
provide a fully rigorous proof under minimal hypotheses, but to explain why
the resulting formula should be true and how it naturally arises from Mellin
inversion and Dirichlet series manipulations.
The
computation illustrates the underlying mechanism: the inverse Mellin transform
produces a Heaviside function that truncates the Dirichlet series, thereby
converting an analytic object into a finite sum. Interchanging the order of
summation and integration, as well as the growth and convergence conditions
required to justify this step rigorously, are not addressed here.
Thus, this
derivation should be read as a heuristic roadmap. It guides the reader toward
the correct statement and clarifies the analytic structure behind it, leaving
the full justification to later sections or to the standard machinery of
analytic number theory. For the purposes of this book, this level of
explanation is both sufficient and instructive.
Main topic: The technical
details, avoid the interchange of the order of summation and integration. Let’s
see how.
Consider the
series
with abscissa
of convergence and 
. Perron’s formula holds for the
first sum because, in the case of a finite sum, the integral may be brought
inside the summation. Consequently, it is enough to prove that the tail sum
vanishes as 
grows.
Take Abel’s
summation formula (see The Riemann Hypothesis Revealed: A path from Numbers to Asymptotic Summation) for the second sum with 
, 
and 
.
Set
Clearly
Hence
But we can find 
, 
such that
From this we get
which tends to
for 
.
Since 
and 
we just need to ensure
that 
.
Therefore, to be safe,
This bound
will be crucial in what follows.
Now assume 
.
To estimate
for 
we fix a large 
and compute
This sum is equal to
according to Cauchy’s theorem.
We have 
because 
.
Change the variable to 
so
Recall that for
Now estimate 
where 
.
We have 
so
Thus,
and
But , and 
, it follows that 
. Therefore, we may continue with the estimate
Choose 
. Then as 
so the whole quantity tends to
Because , the expression vanishes as becomes large.
Therefore,
A similar estimate holds for
It remains to verify that the same
conclusion applies to
We have 
and the integral is equal
to
Now again
Hence
Recall that 
, so
Thus,
Because 
, this expression likewise vanishes in the limit.
The proof of Perron’s formula is now
complete.
Epilogue — Season 2, Episode 9
We end this episode having closed the
contour, controlled the tails, and fixed a small—but instructive—misstep along
the way. As often happens in analytic work, the argument only settles once
every estimate is pushed to its limit and every assumption made explicit.
Episode 9 stands as a reminder that progress in mathematics is rarely
linear: refinement, correction, and clarity are part of the process.
If you enjoyed this deep dive, there’s
much more waiting for you in the blog.
These posts are designed to complement a full, rigorous treatment in the
book—where results like Perron's formula are developed in complete detail, step
by step.
Stay tuned and explore further—this is just a glimpse of the ideas that unfold
in the full text.
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