A Small Taste from My New Book: Season 2 Episode 7

 

Mathematicians sometimes overlook the fact that their audience may not share the same familiarity with the terminology and conventions used in a lecture or written exposition. This is particularly challenging for readers encountering a topic for the first time, or for those who are not accustomed to the author’s personal notation and style. As a result, even well-presented ideas can become difficult to follow if the foundational language is not made explicit.

To avoid unnecessary confusion and to make the discussion as accessible as possible, we will begin by carefully introducing the basic concepts, notation, and assumptions that will be used throughout this text. Establishing this common ground will allow us to focus on the mathematical ideas themselves, rather than on decoding terminology, and will provide a clear baseline from which more advanced material can be developed.

In “The Riemann Hypothesis Revealed” at the section “The growth notation” I explain the idea behind Big‑O notation and why it is useful for describing asymptotic behavior. Specifically:

If  for all , we write

and read “ is big oh of ” to mean that the quotient  is bounded for ; that is, there exists a constant  such that

 for all

 represents the upper bound of the growth rate of a function. It means that, for sufficiently large input sizes, i.e., for all  the function  will not grow faster than a constant multiple of .

Every time we write , the reader should automatically recall the definition above.

The real-variable definition relies on an ordering . That does not exist in  so we cannot use inequalities like  directly. Instead, for two complex functions  we define

as , if there exist constants  and a neighborhood of  such that

for all  in that neighborhood (excluding possibly  itself).

Important nuance: if   we need to consider the direction. Behavior may differ depending on direction in  so typically we specify  uniformly in all directions unless stated otherwise. Of course, direction may be constrained if we work inside a sector or the half-plane.

For example,

1: let

Then

For large  this is bounded (say ), so:

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2:  is .

Indeed, there exists a constant  such that

for all sufficient large .

However,

3:  is  for ,   because

4: A term like  is also  as  grows. Obviously, this belongs to the real case since for  we have

Moreover,

5: combining the above arguments we obtain for  that

6: and if we ask how   behaves for large , where  with  fixed in ?

We have

(easily verified, readers exercise)

and

For large

 lies close to , slightly tilted by .

You can easily verify that as  increases (say )

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With the preliminaries out of the way, it’s time to turn to the real mathematics.

Task: Prove that given ,   and

and then

Work:

In “The Riemann Hypothesis Revealed” (see Logarithmic derivative & Functional formula for Gamma function) I establish the formula

From this we get

Since (see The Riemann Hypothesis Revealed, Asymptotic formulas: Euler’s constant & Harmonic series)

Now for

Thus

Furthermore, by the Euler-Maclaurin summation formula (see A Path from Numbers to Asymptotic Summation in The Riemann Hypothesis Revealed)

for  and

Therefore

Letting

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Corollary: From this we get a less precise estimate for

since .

Furthermore, since  is  for  the estimate reduces to

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Main Topic: In Episode 3 of Season 2 we established the formula:

where

and the sum runs over the nontrivial zeros  of .

Clearly

Let’s see what we can do with the term

Apply

with ,  for . Then,

hence

because as  grows  eventually will dominate every constant.

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Let’s apply this in the formula:

The result is:

This estimate holds uniformly for , with all constants absorbed into the  term, which is dominated by .

Applying this at  we get

Then

because

In Season 2 Episode 4 we established the identity

For  the Dirichlet series on the right-hand side converges absolutely Here  denotes the von Mangoldt function, introduced in The Riemann Hypothesis Revealed and in many standard texts on number theory, and studied extensively in Episode 5 of Season 2.

At ,

Take absolute values:

The series  converges since  and  converges. Indeed, for sufficiently large  we have   so the series is dominated by . More generally one has the standard bound  for any , a very useful asymptotic result but for our purposes it suffices to take .

Thus

From this we get

Let’s split the index set of zeros into “far” and “near”.

where .

The “far” part:

In Season 2 Episode 6 we proved that

The quantity , where  counts the zeros lying inside the rectangle , measures the local density of zeros.

For each integer  the number of zeros with

is ()

Moreover, for zeros with

hence

Thus

because

and

The “near” part:

Here’s why:

For the zeros with , we have the exact equality

The set  contains only finitely many zeros. Additionally,  hence

By the zero-counting estimate  the number of zeros with ordinates in  is .

Thus

Putting everything together we have the estimate

where ,  is a root of zeta in the critical strip,  and .

 

That’s enough for today. None of this is easy—but it’s exactly the kind of structure we can build on top of The Riemann Hypothesis Revealed. Every layer we add makes the landscape a little less mysterious and a little more navigable.

We’ve just walked through one of the most delicate pieces of analytic number theory: controlling the logarithmic derivative of the zeta function by isolating the influence of nearby zeros and showing that everything else collapses into a manageable  term.

This is not bookkeeping. It’s architecture.

• You learned how the zero-counting function  quietly governs the analytic behavior of .
• You saw how the zeros arrange themselves in vertical “shells” around height , and how each shell contributes a controlled amount.
• You discovered why the nearby zeros dominate the local behavior, while the distant ones fade into a logarithmic haze.
• And you used nothing more exotic than careful inequalities, the argument principle, and the Riemann–von Mangoldt formula.

 

This is exactly the kind of scaffolding that The Riemann Hypothesis Revealed gestures toward but doesn’t fully unpack. We’re now filling in the analytic muscle behind the narrative bones.

What we’ve built today is a tool we’ll use again and again:

It’s one of the cleanest windows we have into how the zeros shape the zeta function. And every time you return to it, you’ll see a little more of the underlying geometry.

We’re not just following the book anymore, we’re extending it.

 

P.S. The main reference for this discussion is Multiplicative Number Theory: I. Classical Theory by Hugh L. Montgomery and Robert C. Vaughan.