This exploration delves into the Riemann Zeta Function (RZf), revealing a hidden symmetry in the distribution of prime numbers that provides a framework for understanding its properties. This framework, rooted in the elegant concept of Inverse Prime Symmetry, offers a fresh perspective on the Riemann Hypothesis.

**1. Foundation: Inverse Prime Symmetry**

At the heart of this exploration lies a simple yet profound observation about prime numbers, encapsulated in the following theorem:

**Theorem (Inverse Prime Symmetry):**

Let A = {6x + 5 | x ∈ ℤ} and B = {6y + 7 | y ∈ ℤ} be sets of integers. For any prime number p greater than 3:

- If p ∈ A, then -p ∈ B.
- If p ∈ B, then -p ∈ A.

**Proof:**

**Prime Representation (6k ± 1):**All prime numbers greater than 3 can be expressed in either the form 6k + 1 or 6k – 1.**Set Membership:**Set A corresponds to the form 6k – 1, and set B corresponds to the form 6k + 1.**Negation and Symmetry:**- If p = 6k – 1 (in set A), then -p = -6k + 1 = 6(-k) + 1, which belongs to set B.
- If p = 6k + 1 (in set B), then -p = -6k – 1 = 6(-k) – 1, which belongs to set A.

This theorem establishes a remarkable symmetry: the primes in sets A and B are mirror images of each other with respect to zero. This inherent symmetry becomes the cornerstone of our exploration of the RZf.

**2. The Symmetrized Zeta Function: A Reflection of Prime Symmetry**

Inspired by the Inverse Prime Symmetry theorem, we define a new function, a “symmetrized” version of the Riemann Zeta function, designed to explicitly capture this prime number symmetry:

`ξAB(s) = ∏p∈A (1 - p^(-s))^(-1) · ∏p∈B (1 - p^(-s))^(-1)`

This function reflects the individual contributions of primes from sets A and B to the traditional Riemann Zeta function, making the A-B symmetry explicit.

**3. The Functional Equation: A Mirror of Symmetry**

A crucial aspect of the classical Riemann Zeta function is its functional equation, which connects its values at s and 1-s, revealing a deep symmetry in its behavior. We conjecture that ξAB(s) similarly exhibits a functional equation that reflects the Inverse Prime Symmetry:

`ξAB(s) = ± ξAB(1-s) (Conjectured)`

Deriving the exact form of this functional equation, including the determination of the ± sign, represents a key challenge and a potential avenue for further research.

**4. The Critical Line: A Line of Symmetry**

The line Re(s) = 1/2, known as the critical line, holds immense significance in the study of the Riemann Zeta function. We anticipate that this line acts as an axis of symmetry for ξAB(s), mirroring the A-B symmetry inherent in its definition. This suggests that the critical line plays a crucial role in capturing and revealing the prime number symmetry embedded within the Riemann Zeta function.

**5. A Geometric Lens: The Mellin Transform**

To delve deeper into ξAB(s), we employ the Mellin transform, a powerful tool that connects summation and integration, offering a geometric perspective on the function. We can express ξAB(s) as:

`ξAB(s) = ∫0^∞ ψAB(x) x^(s-1) dx`

Here, ψAB(x) encapsulates information about the distribution of primes within sets A and B, reflecting their symmetrical nature.

**6. Harmonic Echoes: Fourier Analysis**

Exploiting the evenness of ξAB(1/2 + it) about the critical line, we can expand it as a Fourier cosine series:

`ξAB(1/2 + it) = Σ an cos(t log n)`

The coefficients *a_n* hold the key to understanding the intricate dance between the zeros of ξAB(s) and the distribution of primes in sets A and B. The Fourier analysis provides a way to explore this connection through the lens of harmonic oscillations.

**7. Unveiling Hidden Connections: Zeros and Prime Distribution**

The distribution of the zeros of ξAB(s) is expected to be intricately connected to the distribution of primes within sets A and B. The symmetry in the prime distribution, as reflected in the Inverse Prime Symmetry Theorem, is expected to be mirrored in the distribution of the zeros of this symmetrized Zeta function.

**8. Expanding the Horizon: Analytic Continuation**

Leveraging the inverse prime symmetry, we aim to analytically continue ξAB(s), initially defined for Re(s) > 1, to the entire complex plane. This process should inherently reflect the A-B symmetry and offer deeper insights into the function’s behavior. This analytic continuation would allow us to explore the symmetry in a much wider domain and reveal deeper connections to the distribution of prime numbers.

**9. A New Criterion: Li’s Criterion Analogue**

Li’s criterion provides a compelling connection between the Riemann Hypothesis and the non-negativity of specific sums related to the zeros of the Riemann Zeta function. We aim to formulate an analogous criterion for ξAB(s) that incorporates the A-B symmetry:

`λn = 1/(n-1)! d^n/ds^n [s^n-1 log ξAB(s)]|s=1 > 0 for all n ≥ 1 (Conjectured)`

This criterion, if proven, would establish a direct link between the symmetry in the distribution of primes and the behavior of ξAB(s) on the critical line, offering further insights into the Riemann Hypothesis.

**10. An Explicit Connection: The Explicit Formula**

The explicit formula connects the zeros of the Riemann Zeta function to the prime counting function. Similarly, we seek an explicit formula linking the zeros of ξAB(s) to the distribution of primes within sets A and B:

`ψAB(x) = x - Σρ x^ρ/ρ - log(2π) - 1/2 log(1-x^(-2)) (Conjectured)`

where ρ runs over the non-trivial zeros of ξAB(s). This explicit formula, if derived, would provide a powerful tool for relating the properties of ξAB(s) to the distribution of primes in sets A and B, potentially uncovering new connections between prime number theory and complex analysis.

**Conclusion: A Journey of Discovery**

This exploration, rooted in the elegant Inverse Prime Symmetry, offers a novel and potentially powerful framework for investigating the Riemann Zeta function. By constructing a symmetrized Zeta function, exploring its properties, and drawing parallels to the classical theory, we open up potential avenues for future research.