Algebraic Convolution of Zero Sets: The Star Product and the Critical Line 827

Introduction

The study of the Riemann zeta function and the location of its non-trivial zeros has traditionally relied on the tools of complex analysis and analytic number theory. However, the paper arXiv:hal-02373243 introduces a fundamentally different approach by defining an algebraic operation known as the star product (denoted as ☆). This operation acts on polynomials and entire functions by performing a multiplicative convolution of their zero sets. In this framework, if one function possesses zeros at certain locations and another possesses zeros elsewhere, their star product results in a new function whose zeros are the pairwise products of the original sets.

This algebraic-analytic mechanism provides a novel lens through which to view the Riemann Hypothesis (RH). By representing entire functions through their Weierstrass canonical products, the star product allows for a systematic manipulation of divisors. This is particularly relevant to the spectral interpretation of the zeta function, where zeros are viewed as eigenvalues of a hypothetical operator. The work in arXiv:hal-02373243 suggests that the star product mirrors the behavior of eigenvalues under the tensor product of linear operators, offering a formal algebraic structure that could potentially encode the constraints of the critical line.

This article analyzes the technical foundations of the star product, including its determinantal representations and its interaction with p-adic exponentials. We explore how these structures can be used to model the distribution of zeta zeros and propose research pathways that leverage the algebraic symmetries of the star product to investigate the stability of the critical line Re(s) = 1/2.

Mathematical Background

The core mathematical object in arXiv:hal-02373243 is the star product of entire functions defined by their canonical products. For a function f defined by zeros (alpha_i) and a function g defined by zeros (beta_j), the star product f ☆ g is constructed such that its zeros are (alpha_i multiplied by beta_j). This is expressed through the interaction of Weierstrass primary factors E_rho(z). A key identity from the source states that the product of these factors over two different sets of zeros distributes such that the resulting function is a product over the pairwise combined zero set.

A significant contribution of the paper is the determinantal representation of polynomials. For a polynomial P(X), the paper defines a matrix M_P such that P(X) = det(I - M_P X). This companion-like matrix relates the coefficients of the polynomial directly to its spectral properties. In the context of the zeta function, if we consider a truncated product of its zeros, this matrix representation provides a finite-dimensional approximation of the spectral operators often discussed in the Hilbert-Polya conjecture.

Furthermore, the paper defines a zero-pole algebra where the interaction of singularities is governed by simple rules: zero ☆ zero results in a zero, pole ☆ pole results in a zero, and the interaction between a zero and a pole results in a pole. This calculus allows for the treatment of meromorphic functions, such as the zeta function itself, within a unified algebraic framework. The paper also introduces a p-adic exponential, exp_p(X), which utilizes the Moebius function to create a structural link between the star product and prime-power indices, echoing the Euler product decomposition of L-functions.

Spectral Properties and Zero Distribution

Tensor Semantics and Eigenvalue Multiplication

The determinantal identity P(X) = det(I - M_P X) establishes a deep link between the star product and linear algebra. If the roots of P are alpha_i, they correspond to the reciprocals of the eigenvalues of M_P. When we apply the star product to two polynomials P and Q, the roots of the resulting polynomial are the pairwise products of the original roots. This is exactly the behavior of eigenvalues under the Kronecker product (tensor product) of matrices. Specifically, the eigenvalues of the tensor product of two matrices are the products of the eigenvalues of the individual matrices.

This suggests that the star product is the algebraic shadow of a hidden tensor category. In the study of the Riemann Hypothesis, if the zeros of the zeta function are interpreted as the spectrum of an operator, the star product provides the means to analyze how that spectrum behaves under tensorial iteration. If the zeros of the zeta function can be shown to lie on a unit circle in a transformed coordinate z = exp(s - 1/2), then the star product would preserve this unit-modulus property, as the product of two numbers on the unit circle remains on the unit circle.

The Inversion Identity and Functional Symmetry

The source paper arXiv:hal-02373243 proves that the star product is compatible with inversion: (P(1/X) ☆ Q(1/X)) = (P ☆ Q)(1/X). This algebraic symmetry is a discrete analogue of the functional equation of the Riemann zeta function, which relates ζ(s) to ζ(1-s). In a coordinate system where s = 1/2 + it, this symmetry corresponds to a reflection across the critical line. The fact that the star product commutes with such reciprocal transformations suggests that it is a natural operation for preserving the structural symmetries required by the functional equation.

Prime-Local Filtering via exp_p

The construction of exp_p(X) in the source paper allows for the isolation of specific prime-power coefficients within a power series. By star-multiplying a function f(X) by exp_p(X), one effectively filters the exponential generating series to retain only those terms indexed by powers of p. This operation provides a way to decompose global analytic data into prime-local components. In analytic number theory, this mirrors the decomposition of the zeta function into its local Euler factors. The star product thus offers a method to manipulate the "local" zeros of the zeta function and study their contribution to the global distribution of non-trivial zeros.

Novel Research Pathways

1. Stability of the Critical Line under Star Iteration

A promising research direction involves mapping the critical strip of the zeta function to the unit disk via a conformal transformation, such as z = exp(s - 1/2). In this coordinate, the Riemann Hypothesis is equivalent to the statement that all zeros of the transformed function lie on the unit circle |z| = 1. Using the star product, one could investigate the spectral stability of these zeros. If a function's zeros lie on the unit circle, any star-power of that function will also have zeros on the unit circle. Research could focus on whether the zeta function acts as a fixed point or a generator for a specific class of star-invariant entire functions.

2. Determinantal Dynamics of Truncated Zeta Products

Using the matrix M_P defined in arXiv:hal-02373243, one can construct finite-dimensional approximations of the zeta function's zeros. By examining the trace of the powers of these matrices, which correspond to the Newton sums of the zeros, researchers can apply the star product to combine different segments of the zero set. The goal would be to prove that the positivity of certain symmetric polynomials, which is necessary for the Riemann Hypothesis, is preserved under the star product. This would link the algebraic combinatorics of the source paper to the Li criterion for the Riemann Hypothesis.

Computational Implementation

The following Wolfram Language code demonstrates the star product logic by calculating the pairwise products of roots and visualizing the transformation of zeta zeros in the exponential coordinate system.

(* Section: Star Product of Zeta Zero Truncations *)
(* Purpose: Demonstrate the algebraic convolution of zeros *)

Module[{zetaZeros, transformedZeros, scaleFactor, starredZeros, p1, p2},
  
  (* 1. Obtain first 15 non-trivial zeros of Zeta *)
  zetaZeros = Table[N[ZetaZero[k]], {k, 1, 15}];
  
  (* 2. Map to exponential coordinate z = Exp[s - 1/2] *)
  (* Under RH, these should all have Abs[z] == 1 *)
  transformedZeros = Exp[zetaZeros - 1/2];
  
  (* 3. Define a scaling zero from another function (e.g., 0.9 + 0.1i) *)
  scaleFactor = 0.9 + 0.1*I;
  
  (* 4. Perform the Star Product: multiply roots pairwise *)
  (* Here we multiply the zeta set by our scale factor *)
  starredZeros = Table[z * scaleFactor, {z, transformedZeros}];
  
  (* 5. Visualization *)
  p1 = ListPlot[ReIm /@ transformedZeros, 
    PlotStyle -> {Blue, PointSize[0.02]}, 
    PlotLabel -> "Original Zeta Zeros in Z-plane"];
    
  p2 = ListPlot[ReIm /@ starredZeros, 
    PlotStyle -> {Red, PointSize[0.02]}, 
    PlotLabel -> "Starred Zeros (Scaled)"];
    
  Print[GraphicsGrid[{{p1, p2}}]];
  
  (* 6. Check Modulus Preservation *)
  Print["Mean Modulus of Original Transformed Zeros: ", 
    Mean[Abs[transformedZeros]]];
  Print["Mean Modulus of Starred Zeros: ", 
    Mean[Abs[starredZeros]]];
]

Conclusions

The star product framework presented in arXiv:hal-02373243 provides a robust algebraic foundation for the study of zero distributions. By treating the zeros of the Riemann zeta function as elements of a multiplicative convolution algebra, we gain new tools to analyze the symmetries of the critical line. The determinantal representation and the p-adic filtering mechanisms offer clear pathways to connect prime-number theory with spectral analysis. The most promising avenue for future research lies in investigating the invariance of the critical line under star-product transformations and using these algebraic structures to formulate new criteria for the location of non-trivial zeros.

References

• arXiv:hal-02373243
• Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press.
• Verger-Gaugry, J.-L. (2008). On the Parry Polynomial of a G-number. arXiv:hal-00341880.