The Algebra of Zero Multiplication: Spectral Tensors and the Riemann Critical Line

Introduction

The quest to resolve the Riemann Hypothesis (RH) has long sought a bridge between the analytic properties of the zeta function and an underlying algebraic or spectral structure. Traditional methods focus on the distribution of zeros on the critical line Re(s) = 1/2, yet the search for a linear operator whose eigenvalues correspond to these zeros remains one of the most significant challenges in modern mathematics. The paper arXiv:hal-02373243v1, titled "The Ene-product over a commutative ring," introduces a powerful algebraic operation—the Ene-product—that may offer a new pathway to this goal.

The Ene-product, denoted by the star symbol (☆), is an internal law on formal power series and entire functions that acts directly on their zero sets. Specifically, if two functions possess zeros at specific locations in the complex plane, their Ene-product possesses zeros at the pairwise products of those locations. This property, combined with a rigorous determinantal representation for polynomials and series, aligns the Ene-product with the spectral interpretations of L-functions and the Hilbert-Polya conjecture.

This analysis explores the connection between the Ene-product and the Riemann zeta function. We demonstrate how the algebraic structures defined in arXiv:hal-02373243v1, such as the companion-like matrix M_P and the prime-local filter exp_p, provide tools for isolating prime-power data and manipulating zero distributions. By mapping the critical line into the multiplicative domain of the Ene-product, we identify novel research directions for studying the stability and symmetry of the zeta zero set.

Mathematical Background

The Ene-Product and Zero Multiplication

As established in arXiv:hal-02373243v1, the Ene-product is defined on the set of formal power series with constant term 1. Its most defining characteristic is its effect on the roots of polynomials. For two polynomials f(X) and g(X) with roots (alpha_i) and (beta_j) respectively, the Ene-product (f ☆ g)(X) is a polynomial whose roots are exactly the products (alpha_i * beta_j). This multiplicative behavior extends to entire functions of finite genus through the Weierstrass factorization.

The source paper proves that if f and g are entire functions represented by their Weierstrass factors E_rho(z/alpha_i) and E_rho(z/beta_j), then their Ene-product follows a formal factorization where the resulting factors are E_rho(z / (alpha_i * beta_j)). This transformation is significant because it suggests that algebraic operations on the function itself can be mapped directly to arithmetic operations on the zero set.

Determinantal Representation and Tensor Products

A central theorem in arXiv:hal-02373243v1 provides a determinantal formula for any polynomial P(X) of degree d. It identifies a matrix M_P such that P(X) = det(I - M_P X). More importantly, it demonstrates that the Ene-product of two polynomials corresponds to the tensor product of their associated matrices:

• Linear Representation: P(X) ☆ Q(X) = det(I - (M_P ⊗ M_Q) X).
• Eigenvalue Correspondence: The eigenvalues of the tensor product M_P ⊗ M_Q are precisely the pairwise products of the eigenvalues of M_P and M_Q.

This result is a concrete realization of the idea that functions can be represented by operators whose spectra encode the zeros. In the context of the Riemann Hypothesis, this provides a framework for treating the zeros of the zeta function as eigenvalues of a structured operator.

Main Technical Analysis

Spectral Properties and Zero Distribution

The spectral representation of the Ene-product allows us to re-examine the distribution of non-trivial zeros of the zeta function. If we consider a function whose zeros are mapped from the critical line Re(s) = 1/2 to a unit circle (or a circle of radius R) via an exponential map, the Ene-product acts as a mechanism for additive convolution of the ordinates. Specifically, mapping a zero rho = 1/2 + i*gamma to alpha = exp(c*rho) turns the multiplication of zeros in the Ene-domain into the addition of the imaginary parts gamma.

The tensor product formula implies that the traces of the powers of the companion matrix M_P encode the power sums of the zeros. Since the coefficients of the zeta function and its derivatives are related to these power sums, the Ene-product provides a way to relate the moments of the zeta zeros to the coefficients of a higher-order spectral operator. This allows for the derivation of new constraints on the variance and spacing of zeros, which are central to the GUE (Gaussian Unitary Ensemble) conjecture and the statistical verification of the Riemann Hypothesis.

Algebraic Structures and Prime-Local Filters

The source paper arXiv:hal-02373243v1 introduces a special series denoted as exp_p(X), which functions as a prime-local filter. The identity established is:

exp_p(X) ☆ f(X) = exp(- Sum_{k=1 to infinity} F_(p^k) * X^(p^k))

where f(X) is the exponential of a series with coefficients F_i. This operator effectively isolates or "surges" the prime-power contributions within the generating function. Because the logarithm of the Riemann zeta function is defined by a sum over prime powers (via the von Mangoldt function), the exp_p operator acts as a projection onto the p-adic component of the Euler product.

This "zero/pole arithmetic" (where zero ☆ zero = zero and zero ☆ pole = pole) provides a symbolic calculus for manipulating the singularities of L-functions. By applying successive exp_p filters to a function encoding the zeros of the zeta function, one can theoretically isolate the contribution of individual primes to the global distribution of zeros, offering a new method for exploring the local-to-global transition in analytic number theory.

Novel Research Pathways

Ene-Operator Spectral Theory on the Critical Line

We propose the development of an "Ene-operator" theory where the Riemann zeta function (or the completed xi-function) is treated as the characteristic determinant of an infinite-dimensional operator. The research would focus on identifying the class of operators whose tensor products preserve the property that all eigenvalues have a constant modulus. If the Ene-product of the zeta function with itself maintains a specific growth rate in its coefficients, it would imply that the zeros must remain on the critical line to satisfy the spectral radius constraints of the tensor product.

Sumsets of Ordinates and Multiplicative Stability

A second pathway involves mapping the zeta zeros to the unit circle and analyzing the "multiplicative Minkowski product" of the zero set. By examining the roots of the n-th Ene-power of a truncated zeta polynomial, researchers can study how violations of the Riemann Hypothesis (zeros off the critical line) would amplify under the Ene-product. Off-line zeros would result in roots with moduli that deviate exponentially from the expected value, providing a highly sensitive diagnostic tool for numerical and theoretical investigations of zero clustering.

Computational Implementation

(* Section: Ene-Product and Zeta Zero Analysis *)
(* Purpose: Demonstrates how the Ene-product multiplies mapped zeta zeros *)

Module[{nZeros = 10, gammas, alphas, polyP, polyQ, starPoly, starRoots},
  
  (* 1. Obtain the first n non-trivial zero ordinates of the Zeta function *)
  gammas = Table[Im[ZetaZero[k]], {k, 1, nZeros}];
  
  (* 2. Map ordinates to the unit circle: alpha = exp(i * gamma) *)
  (* This transformation maps the critical line Re(s)=1/2 to |z|=1 *)
  alphas = Exp[I * gammas];
  
  (* 3. Define a function to construct a polynomial from its roots *)
  rootsToPoly[rs_] := Expand[Product[x - r, {r, rs}]];
  
  (* 4. Create the initial polynomial P(x) using the mapped zeros *)
  polyP = rootsToPoly[alphas];
  
  (* 5. Define the Ene-product operation (root multiplication) *)
  (* In a full implementation, this uses the tensor product of matrices *)
  eneStarRoots[rootsA_, rootsB_] := Flatten[Outer[Times, rootsA, rootsB]];
  
  (* 6. Compute the Ene-self-product of the zeta zero polynomial *)
  starRoots = eneStarRoots[alphas, alphas];
  starPoly = rootsToPoly[starRoots];
  
  (* 7. Output results and check magnitudes *)
  Print["Original Mapped Zero Count: ", Length[alphas]];
  Print["Ene-Product Zero Count: ", Length[starRoots]];
  
  (* Visualization of the zero distribution on the complex plane *)
  Print[ListPlot[Table[{Re[r], Im[r]}, {r, starRoots}], 
    PlotLabel -> "Roots of (Zeta-Polynomial) Star (Zeta-Polynomial)",
    AxesLabel -> {"Re", "Im"}, 
    PlotStyle -> PointSize[Medium]]];
    
  (* Diagnostic: check if all roots still lie on the unit circle *)
  Print["Max deviation from unit circle: ", Max[Abs[Abs[starRoots] - 1]]];
]

Conclusions

The Ene-product framework introduced in arXiv:hal-02373243v1 provides a rigorous algebraic foundation for the multiplicative manipulation of zero sets. By establishing that the Ene-product of functions corresponds to the tensor product of their underlying spectral operators, the paper offers a concrete mechanism for exploring the Hilbert-Polya conjecture. The most promising avenue for further research lies in the application of exp_p filters to isolate prime-power data and the use of Ene-powers to amplify and detect deviations from the critical line. These tools provide a sophisticated new language for describing the symmetries of the Riemann zeta function and its non-trivial zeros.

References

• arXiv:hal-02373243v1 - The Ene-product over a commutative ring.
• H. L. Montgomery, "The pair correlation of zeros of the zeta function," Proceedings of Symposia in Pure Mathematics, 1973.
• E. C. Titchmarsh, "The Theory of the Riemann Zeta-Function," Oxford University Press.