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.