A Small Taste from My New Book: Season 2 Episode 6

 

In this episode, we step from global statements about the zeros of the Riemann zeta function into their local behavior. While classical results describe how many zeros lie below a given height on average, they say very little about what happens in short vertical intervals. Yet it is precisely this local structure—clustering, spacing, and short-scale irregularity—that reveals the true analytic complexity of the zeta function.

To address this, we return to one of the most versatile tools in complex analysis: Jensen’s formula. By carefully choosing the center and radius of the Jensen disk, and by exploiting regions where the zeta function (or its completed form ξ) is well-controlled, we turn analytic growth estimates into concrete bounds on the number of zeros in narrow strips.

The geometry here is not incidental. The specific placement of the disk, the choice of radii, and the alignment with the interval  are all tuned to capture exactly the zeros that matter—no more, no less. This episode explains why these choices work, how symmetry enters the picture, and how local zero density can be bounded using purely analytic means.

What emerges is a clear principle: local zero counts are governed not by global asymptotics, but by precise analytic control in carefully chosen regions.

 In “The Riemann Hypothesis Revealed” we proved the standard Jensen’s formula (centered at ).

The formula requires  to be analytic in , . If  are the zeros of  in  then

This follows from the fact that  is harmonic except at zeros and the Poisson integral formula.

Now we shift the center.  Suppose we want Jensen’ formula around .

Let  be analytic in . Define the translated function .

Then

-  is analytic in  

- and

Zeros transform as

Thus

where  are zeros of .

Apply Jensen to

Substitute back  and  to get Jensen’s formula centered at

where  analytic in ,  and  zeros of .

The formula states:

Value at center = average on boundary – zero contribution

which comes from the subharmonic nature of  in Complex Analysis.

▪▪▪

In classical proofs controlling zeros of the Riemann zeta function  or its completed function , we use  and  and center Jensen near the line . The choice is not arbitrary. It’s carefully tuned so that:

- the circle of integration intersects the critical strip, and

- the function is analytic in a slightly larger disc.

Let me show the geometric reason.

We center Jensen at . This point lies safely to the right of the critical strip. At  we have strong bounds for  because the Dirichlet series converges absolutely.

The circle used in Jensen is . Let . Then . The smallest real part occurs at , . Thus, the circle reaches . The disk covers

Therefore, every zero satisfying  must lie in .

We assume analyticity in the larger disk . Note . This extra margin allows estimates on the boundary when using

- Maximum modulus principle

- bounds on  for .

The smallest real part in the larger disk is . Thus, analyticity is required only for  which holds for the entire function  or for suitably modified zeta expressions.

What Jensen then gives:

The left side measures how many zeros lie in the disk (see The Riemann Hypothesis Revealed). Bounding the right side yields an upper bound on zero density near height .

and

Hence,

Since  is larger than every  , let  denote the number of roots  such that . Then, following the proof from The Riemann Hypothesis Revealed,

Thus,

By the way  and .

Why the center has .

The shift  places the disk () symmetrically around the segment

Indeed, we take

For  

This interval contains the height  but avoids some boundary issues when averaging.

The interval of interest for zero counting is . Its midpoint is . So choosing  means the disk is centered exactly at the midpoint of the vertical interval we care about.

The key idea is the numbers  are chosen so that

- the circle reaches deep into the critical strip

- the center lies where zeta is easily bounded

- the disk still stays inside a region where the function is analytic.

- The disk  intersects the strip

- The segment  sits nicely inside the vertical span of the disk.

- The disk extends beyond it by equal margins (bellow: , above: )

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

In contrast, global results such as the asymptotic formula for , describe only the total number of zeros up to height  . While useful, these global asymptotics do not rule out extreme local behavior, such as large clusters of zeros in short intervals or unusually large gaps. To understand such local phenomena, it is therefore essential to study the increment  .

By choosing a disk that fully covers the right half of the critical strip over the interval , we obtain the bound

This estimate will be combined with the fundamental symmetry of the zeros of the Riemann zeta function: if  is a zero, then so is  (see The Riemann Hypothesis Revealed). As a consequence, the zeros occur in symmetric pairs about the critical line, and we may write

Our goal is to show that the number of zeros in any unit-height interval grows at most logarithmically. More precisely, we will prove that

for the zeros of the Riemann zeta function. The proof is based on Jensen’s formula, applied to the completed zeta function .

We have

hence

For the pi-factor

Since , .

In “The Riemann Hypothesis Revealed”, my book, I prove that the growth of zeta inside the critical strip is

for all . However,  behaves badly near  so we can take

Obviously, for all , . So:  and  which allows:

The choice of  is to keep the logarithm positive for small values of .

For

So:

Taking logarithms

Absorbing the constants (), for , one has the classical bound:

Now, for Gamma, in “The Riemann Hypothesis Revealed” we established a crude estimate

Observe that for

This means that along vertical lines gamma is .

We can do better. Away from the real axis we apply Stirling’s formula

Then

On vertical lines  so . Also . Thus,

and

From this we get the sharp

because the term  is  and constants like  are absorbed into .

Now we have all the ingredients to derive a standard growth bound for the Riemann xi function.

Let’s put everything together cleanly.

Start from the identity

Take logarithms (this is the cleanest way):

Insert the bounds

Zeta

Gamma

Pi-factor

Polynomial factors

for fixed , since

Combine everything

Exponentiating:

for some constant .

In Jensen we need to control  from above. Having

then certainly

Now observe:

On disk , we have

So uniformly:

Also:

Thus:

Therefore,

Epilogue

This episode shows how a classical identity—Jensen’s formula—becomes a powerful microscope when applied with geometric insight. By centering the analysis at the right point and working within a disk that intersects the critical strip in a controlled way, we obtain a sharp bound on the number of zeros in a unit-height interval. The result,

is not merely technical—it is a statement about how wildly (or rather, how tamely) zeros are allowed to fluctuate locally.

Equally important is the role of symmetry. The pairing  is not an aesthetic curiosity; it is what converts a bound in a disk into a statement about zeros on one side of the critical line. Analytic growth, geometric placement, and functional symmetry work together here with remarkable efficiency.

In the next episode, we will push these ideas further. Having gained control over local zero density, we will begin to connect these bounds to deeper structural questions about zero distribution, spacing, and their impact on explicit formulas. The machinery is now in place—the next step is to see what more it can reveal.

This section continues. Stay tuned.

 

P.S. In software engineering, integration tests verify that individual components don’t just work in isolation but collaborate correctly as a coherent system. They expose subtle faults that only appear when ideas, interfaces, and assumptions meet. This blog does exactly that kind of deep integration testing for my books The Riemann Hypothesis Revealed and The Essential Transform Toolkit: concepts are not merely presented, but forced to interact, cross‑validate, and reinforce one another. What emerges is not a collection of standalone results, but a unified mathematical narrative whose strength lies precisely in how well its parts fit together.