Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erdős--Straus Conjecture, Subject to its Analytic Continuation

1. Problem Statement

The paper addresses two distinct but profound problems in number theory:

1. The Generalized Erdős–Straus Conjecture: This conjecture posits that for any integer $n \ge 2$ and fixed integer $k \ge 4$, the equation $\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has solutions in positive integers $x, y, z$. The specific case $k=4$ is the classical Erdős–Straus conjecture.
2. The Riemann Hypothesis (RH): The conjecture that all non-trivial zeros of the Riemann zeta function $\zeta(s)$ lie on the critical line $\Re(s) = 1/2$.

The Core Question: Can the algebraic structure of the Erdős–Straus decomposition, specifically the polynomial function $F$ used to generate solutions, be analytically continued to the complex plane to reveal a structural connection to the Riemann zeta function and its zeros?

Note: The author explicitly states that this paper does not claim to prove the Riemann Hypothesis but rather proposes a formal framework to explore a potential link.

2. Methodology

The author employs a method of parametric extension and analytic continuation:

• Step 1: Algebraic Reformulation of the Conjecture.
  Building on previous work, the author utilizes a quadratic parametrization. The equation $\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is shown to be equivalent to the existence of integers $x, t$ such that the function:
  $$F^{(k)}_{x,t}(n) = t^2(kx - n)^2 - 2nxt$$
  is a perfect square ($m^2$). If this condition holds, $y$ and $z$ are derived as roots of a quadratic equation.

• Step 2: Real Parametric Extension.
  The integer $n$ is replaced by a real variable $n^s$ (where $s > 1$), and the parameters $x, t$ are allowed to be real numbers. The function becomes:
  $$F^{(k)}_{x,t}(n^s) = t^2(kx - n^s)^2 - 2n^sxt$$
  The author demonstrates that for any $n \ge 1$ and $s > 1$, one can always choose real parameters $x, t$ such that $F^{(k)}_{x,t}(n^s)$ is a perfect square of a real number $m_s(n)$.

• Step 3: Decomposition and Summation.
  Using the condition that $F$ is a perfect square, the author defines real numbers $x_s(n), y_s(n), z_s(n)$ such that:
  $$\frac{k}{n^s} = \frac{1}{x_s(n)} + \frac{1}{y_s(n)} + \frac{1}{z_s(n)}$$
  Summing this equality over all integers $n \ge 1$ yields:
  $$\sum_{n=1}^{\infty} \frac{k}{n^s} = \sum_{n=1}^{\infty} \left( \frac{1}{x_s(n)} + \frac{1}{y_s(n)} + \frac{1}{z_s(n)} \right)$$
  The left-hand side is identified as $k\zeta(s)$.

• Step 4: Hypothesis of Analytic Continuation.
  The author defines a new function $G_k(s)$ as the sum on the right-hand side. The central hypothesis is that if the quantities $x_s(n), y_s(n), z_s(n)$ (derived from the roots of the quadratic equation associated with $F$) admit an analytic continuation to the complex plane $\mathbb{C}$, then $G_k(s)$ becomes a meromorphic function satisfying $G_k(s) = k\zeta(s)$ for all $s \in \mathbb{C} \setminus \{1\}$.

3. Key Contributions

• Formal Equivalence: Establishes a formal identity $G_k(s) = k\zeta(s)$ for $\Re(s) > 1$ by decomposing the term $k/n^s$ into a sum of three reciprocal terms derived from the Erdős–Straus polynomial structure.
• Quadratic Symmetry: Identifies that the terms $y_s(n)$ and $z_s(n)$ are the roots of the quadratic equation:
  $$V^2 - 2t(kx - n^s)V + 2n^sxt = 0$$
  The author highlights a striking symmetry: these roots are symmetric with respect to the value $t(kx - n^s)$, analogous to the symmetry of the non-trivial zeros of $\zeta(s)$ with respect to the critical line $\Re(s) = 1/2$.
• Functional Equation Inheritance: Argues that if the analytic continuation holds, $G_k(s)$ inherits the functional equation of the Riemann zeta function, suggesting that the properties of the polynomial $F$ and its roots directly dictate the properties of $\zeta(s)$.

4. Results

• Theorem 1: For real $s > 1$, the function $G_k(s)$ defined by the sum of the reciprocals of the parametrized roots is formally equal to $k\zeta(s)$.
• Conditional Corollary: If the functions $x_s(n), y_s(n), z_s(n)$ can be analytically continued to complex $s$, then $G_k(s) = k\zeta(s)$ holds for all complex $s$ (except the pole at $s=1$). Consequently, the zeros of $G_k(s)$ would coincide exactly with the zeros of $\zeta(s)$.
• Structural Insight: The paper posits that the "zeros" of the polynomial $F$ (where $y_s(n) = z_s(n)$) and the behavior of the quadratic roots as functions of $s$ are intrinsically linked to the location of the Riemann zeros.

5. Significance and Future Directions

• Novel Perspective: The paper proposes a radical shift in viewing the Riemann Hypothesis: instead of analyzing $\zeta(s)$ directly, one might analyze the algebraic properties of the polynomial $F^{(k)}_{x,t}(n^s)$ and its roots.
• Bridging Discrete and Continuous: It attempts to bridge the gap between discrete number theory (Diophantine equations like Erdős–Straus) and complex analysis (Riemann zeta function) through a unified parametric framework.
• Open Questions for Specialists: The author explicitly defers the rigorous proof of the analytic continuation to specialists in complex analysis. Key open questions include:
  • Can $x_s(n), y_s(n), z_s(n)$ be defined as meromorphic functions of $s$?
  • Does the symmetry of the quadratic roots imply the critical line location of $\zeta(s)$ zeros?
  • Can numerical exploration of $F(n^s)$ on the critical line provide clues to the distribution of zeros?

Conclusion:
The paper does not solve the Riemann Hypothesis but offers a heuristic and formal framework suggesting that the structure of the generalized Erdős–Straus conjecture, when extended to the complex domain, generates the Riemann zeta function. If the proposed analytic continuation is valid, the study of the polynomial $F$ becomes a direct pathway to understanding the zeros of $\zeta(s)$. The author calls for rigorous investigation by experts in complex analysis to validate the existence of the function $G_k(s)$ and its properties.