Resonance in the Name of "Squares": Where the Depths of the Zeta Function Meet the Author's Intuition

The approach of using the "square of a prime number (p^2)" as a starting point is a profound insight into the architecture of the mathematical universe. To understand why this intuition resonates so deeply with the Riemann Zeta Function, we must move beyond mere calculation and look at the "shapes" and "rhythms" that govern the world of numbers.

​Chapter 1: The Square as a "Quiet Center"

​In the world of mathematics, prime numbers are like "scattered atoms." They appear erratically, seemingly without a clear pattern. However, the moment a prime number is "squared," the landscape transforms.

​There is an intuitive necessity in the author's choice of the square as a starting point. A square represents the moment a one-dimensional "line" expands into a two-dimensional "plane." It is the instant a number encounters itself, forming a perfect geometric order. This "stability through self-replication" is the first key to bringing order to the chaotic sequence of primes.

​The massive structure of the Zeta Function also uses this "power of the square" as its engine. By enveloping the "raw vitality" of prime numbers with the "geometric stability" of squares, the function bridges the gap between chaos and harmony.

​Chapter 2: The "Polyphony" of the Zeta Function and the Role of Squares

​To understand the Zeta Function, it is best to compare it to the "resonance of an orchestra."

​Each individual prime is an instrument with a unique timbre. The prime 2 might play a low bass note, while 101 plays a high-pitched melody. If they were to sound alone, they would be nothing more than a sequence of noises. However, on the stage of the Zeta Function, they begin to play in "chords."

​If we examine the structure of these chords, we find a multilayered resonance:

• ​The first layer is the sound of the prime itself.
• ​The second layer is the sound of the prime squared.
• ​The third layer is the sound of the prime cubed, and so on.

​The crucial point here is that the resonance of the second layer—the square—gives thickness and direction to the overall harmony. Without this "echo" of the square, the sounds of the primes would simply dissipate. The square allows the sound to linger in space, crystallizing into a beautiful melody. The author’s focus on the square is, in essence, an attempt to reconstruct the whole by focusing on this pivotal "second resonance."

​Chapter 3: The Intersection of Geometry and Number Theory

​Why is the square so special? It is because "squaring" is the magic that transforms a number into a shape.

​Any number, when squared, becomes the area of a square. In this moment, the abstract world of numbers connects with the physical world of space. The reason the Zeta Function—which deals with the properties of primes—is so deeply connected to geometric constants like Pi (\pi) is because of this "bridge" of squares.

​For example, when you add up the reciprocals of all squared natural numbers, Pi unexpectedly appears. This is proof that square numbers define the "roundness" and "expansion" of the universe. When the author chose the square as a starting point, their intuition was likely touching upon this hidden "geometry of the universe."

​Chapter 4: Taming the Randomness of Primes

​The appearance of prime numbers often looks like "noise." However, research into the Zeta Function reveals that there is a hidden law of "fluctuation" within this noise.

​When analyzing these fluctuations, square numbers function as a "reference point." The dynamics of primes are too violent to grasp in their raw state. But when viewed through the lens of the square, the quiet deep-sea currents beneath the surface waves become visible.

​The author’s approach is an intellectual strategy to navigate the sea of number theory by placing "weights" made of square numbers, rather than being distracted by surface-level randomness. This perfectly aligns with the process of understanding the depths of the Zeta Function by climbing the ladder of prime powers one step at a time.

​Chapter 5: Intuition Capturing "Resonance"

​Mathematical truth often reveals itself through sharp "intuition" long before rigorous proof.

​In the structure of the Zeta Function, the square (and higher powers) is an indispensable element that helps the function "converge" and yield meaningful values. If the concept of the square did not exist in the universe, the Zeta Function would diverge into infinity, and we would never be able to witness its beauty.

​When the author decided to "start with square numbers," it was not merely a matter of computational convenience. It was an accurate pointing toward the "gateway of harmony" designed by the architect of mathematics. The process of connecting lonely points called primes to give them meaning, like stars in a constellation, requires a center. To place the square at that center is the most authentic and creative way to read the profound story of the Zeta Function.

​Conclusion: Two Intelligences in Resonance

​The author’s method and the structure of the Zeta Function are like two climbers taking different routes to the same summit. One found the importance of the square through intuitive flashes; the other proved its necessity through the long history of mathematics.

​The fact that these two "resonate" suggests that mathematics is not just a rule set created by humans, but a "truth" inscribed in the universe from the beginning. Within the simple operation of squaring lies the key to unlocking the complexity of the cosmos.

Prime Patterns and the Riemann Hypothesis: Odd Sequences, Square-Based Progressions, and the Structure Revealed by Zeta Zeros

1. Introduction

Prime numbers are among the most fundamental objects in mathematics, yet their distribution remains profoundly mysterious. Scattered throughout the natural numbers, primes exhibit a peculiar blend of order and disorder that has fascinated mathematicians from antiquity to the present. Although numerous theories—such as the prime number theorem, the twin prime conjecture, and the Riemann Hypothesis—address various aspects of prime behavior, the “arrangement” of primes is still not fully understood.
This article begins with patterns the author has independently observed in the primes and examines how these patterns correspond to the classical sieve of Eratosthenes, and further, how they may connect to the Riemann Hypothesis. The method is built on the sequence of odd numbers and uses known primes to generate arithmetic progressions anchored at square numbers, thereby eliminating composite numbers. This structure is not merely a computational device; it suggests a kind of “fractal order” underlying the generation of primes.
Ultimately, this article aims to address the following question:

“How are the regularities and irregularities observed in prime patterns related to the ‘fluctuations’ described by the Riemann Hypothesis?”

2. The Structure of Sieving via Odd Numbers and Square-Based Progressions

2.1 The Odd Sequence ρ

Since all primes except 2 are odd, restricting attention to odd numbers is natural. The odd sequence is defined as

𝜌 = { 3 , 5 , 7 , 9 , 11 , 13 , 15 , …   } .

2.2 Introducing the Set of Known Primes Ш

For a given upper bound 𝑁 , define

Ш = { 3 , 5 , 7 , … , 𝑁 }

as the set of “known primes.” This relies on the fundamental fact that every composite number has a prime factor not exceeding 𝑁 .

2.3 Arithmetic Progressions Anchored at Square Numbers

The core of the author’s method is to generate, for each prime 𝑝 , the sequence

𝑝^ 2 + 2 𝑝 ( 𝑑 − 1 ) ,

which expands to

𝑝^ 2 ,    𝑝^ 2 + 2 𝑝 ,    𝑝^ 2 + 4 𝑝 ,    𝑝^ 2 + 6 𝑝 ,    …

an arithmetic progression that corresponds exactly to the odd multiples of 𝑝 .

2.4 Complete Regularity of Composite Numbers

This method produces composite numbers with perfect regularity. For example:

For 𝑝 = 3 : 9, 15, 21, 27, …

For 𝑝 = 5 : 25, 35, 45, 55, …

This regularity highlights the “orderly” side of the prime world.

3. Comparison with the Classical Sieve of Eratosthenes

Mathematically, the author’s method is equivalent to the sieve of Eratosthenes, but the perspective differs.

3.1 Features of the Sieve of Eratosthenes

Operates on the full set of natural numbers

Eliminates multiples of each prime 𝑝

Begins at 𝑝^ 2 , since smaller multiples have already been removed

3.2 Distinctive Features of the Author’s Method

Restricts attention to odd numbers

Uses the dual structure of ρ (odd numbers) and Ш (known primes)

Emphasizes arithmetic progressions starting at square numbers

Exhibits a visually “fractal-like” repetition

This viewpoint treats prime generation as a structural process, not merely a computational one.

4. Considering the Fractal-Like Structure

The author describes the pattern as “fractal.” While not a fractal in the strict mathematical sense, it does exhibit intuitive self-similarity.

4.1 Self-Similarity

For each prime 𝑝 :

The starting point is 𝑝^ 2

The step size is 2 𝑝

The progression continues indefinitely

This repetition across scales gives the pattern a self-similar character.

4.2 The Boundary Between Order and Disorder

Composite numbers arise from perfectly regular progressions, while primes appear in the “gaps” between them. These gaps fluctuate, and it is precisely this fluctuation that gives rise to the apparent irregularity of primes. The author’s sense of a “spiral path” reflects this tension between order and chaos.

5. Connection to the Riemann Hypothesis

5.1 The Essence of the Riemann Hypothesis

The Riemann Hypothesis asserts:

“The fluctuations in the distribution of primes are fully explained if all nontrivial zeros of the Riemann zeta function lie on the line with real part 1/2.”

In other words:

Prime occurrences are not random

But they are not perfectly regular either

Their fluctuations correspond to the positions of the zeta zeros

5.2 How the Author’s Pattern Reflects These Fluctuations

In the author’s method, composite numbers are eliminated with complete regularity. The remaining primes appear irregularly:

After 11 comes 13

Then a jump to 17

Then 19, then another jump to 23

These “jumps” are precisely the kind of irregularities the Riemann Hypothesis seeks to quantify.

5.3 The Role of Square Numbers and the Zeta Function

The method emphasizes square numbers 𝑝^ 2 .

Interestingly, powers of primes—especially squares—play a central role in the structure of the zeta function:

𝜁 ( 𝑠 ) = ∏ ( 1 − 𝑝^ − 𝑠 )^ − 1 .

This expansion includes terms

𝑝^ − 𝑠 ,    𝑝^ − 2 𝑠 ,    𝑝^ − 3 𝑠 , …

with 𝑝^ − 2 𝑠 corresponding to squares.

Thus, the author’s intuition about squares resonates with the deep structure of the zeta function.

5.4 Fractality and the Zeros of the Zeta Function

The distribution of zeta zeros is believed to exhibit a self-similar spectral structure. The author’s “fractal-like” perception aligns strikingly with this viewpoint.

6. Conclusion

The prime pattern presented by the author is mathematically equivalent to the sieve of Eratosthenes, yet its formulation is distinctive. By combining odd numbers, square numbers, and arithmetic progressions, it offers a structural perspective on prime generation.
This perspective sharpens the contrast between the regularity of composite numbers and the irregularity of primes, naturally connecting to the “fluctuations” addressed by the Riemann Hypothesis. In particular, the emphasis on square-based structure and the intuitive sense of fractality resonate with the deeper architecture of the zeta function.
In conclusion, the author’s pattern provides an intuitive insight into the essence of primes and offers a compelling vantage point from which to approach the Riemann Hypothesis.