Spectral Invariants of the Critical Line: The Calculus of Superzeta Functions

Executive Summary

The research presented in arXiv:cea-01076225v1_document_document establishes a comprehensive analytic framework for "superzeta" functions, which are Dirichlet-type series where the summation is performed over the non-trivial zeros of the Riemann zeta function. The central insight is the discovery of a systematic dictionary relating the special values of these functions to the logarithmic derivatives of the completed zeta function, Ξ(s). By classifying these functions into three distinct kinds, the source paper provides a unified theory of their analytic continuation and residue structures. This approach is highly promising for the Riemann Hypothesis (RH) because it translates the discrete distribution of zeros into a continuous landscape of analytic invariants. Specifically, the connection to Li-Keiper coefficients allows for a spectral interpretation of the zeros' distribution, where RH is equivalent to the non-negativity of specific superzeta-derived sequences.

Introduction

The distribution of the non-trivial zeros of the Riemann zeta function, typically denoted as ρ = σ + iγ, remains the most significant challenge in analytic number theory. Traditional approaches often rely on the prime number theorem or local statistics of zeros. However, the source paper arXiv:cea-01076225v1_document_document shifts the focus toward global invariants called superzeta functions. These functions, defined as Z(s|t) = ∑ (ρ - (1/2 + t))^-s, treat the zeros as a spectrum of a hypothetical operator.

This methodology traces its roots back to Riemann's unpublished notes and the early 20th-century work of Mellin, but it has been significantly modernized by the inclusion of zeta-regularized determinants and the Li positivity criterion. The paper identifies a remarkable duality: superzeta values at negative integers are rational expressions involving Bernoulli polynomials, while values at positive integers are transcendental, involving the derivatives of log Ξ(s). This article explores how these structures provide a rigorous probe into the critical line and identifies new pathways for computational and theoretical investigation.

Mathematical Background

The foundation of the superzeta theory rests on the completed Riemann zeta function, Ξ(s), which is an entire function satisfying the functional equation Ξ(s) = Ξ(1-s). It is defined as Ξ(s) = (1/2) s (s-1) π^-s/2 Γ(s/2) ζ(s). The zeros of Ξ(s) are exactly the non-trivial zeros ρ of ζ(s).

The source paper defines three primary families of superzeta functions:

• First Kind: Z(s|t) = ∑_ρ (1/2 + t - ρ)^-s, where t is a complex shift parameter.
• Second Kind: 𝒵(σ|t) = ∑_k=1 (τ_k² + t²)^-σ, where τ_k represents the imaginary parts of the zeros.
• Third Kind: A symmetric variant Z(s|τ) combining features of the first two families.

A critical observation in the paper is the comparison with the Hurwitz zeta function. Just as the Hurwitz zeta function ζ(s, w) relates integers to the Γ function, the superzeta function relates Riemann zeros to the Ξ function. This "zeta function of a zeta function" behavior suggests that the zeros follow an algebraic regularity that can be exploited through residue calculus and contour integration.

Main Technical Analysis

Transcendental Identities and Logarithmic Derivatives

The most significant technical contribution of arXiv:cea-01076225v1_document_document is the explicit evaluation of Z(n|t) for positive integers n. For n > 1, the paper proves the identity: Z(n|t) = ((-1)^n-1 / (n-1)!) (log Ξ)⁽ⁿ⁾(1/2 + t).

This identity links the sum of powers of the zeros directly to the n-th derivative of the logarithm of the completed zeta function. This is vital for the Riemann Hypothesis because the Keiper-Li coefficients λ_n—whose non-negativity is equivalent to RH—are themselves linear combinations of these logarithmic derivatives. Consequently, the superzeta function acts as a generating function for the very coefficients that encode the truth of the critical line hypothesis.

The Rational-Transcendental Duality

The paper exhaustively tabulates values for Z(s|t) at integer points. At negative integers s = -n < 0, the values are purely rational, often involving Bernoulli polynomials B_n(w). For example, the paper shows that at s = -n, the value is related to -(1/(n+1)) B_n+1(w). This suggests that the zero set, despite its apparent complexity, possesses a hidden arithmetic structure. Conversely, at s = 0, the derivative Z'(0|t) provides the zeta-regularized product of the zeros, which is shown to be essentially log Ξ(1/2 + t). This identifies Ξ(s) as the spectral determinant of the zero set.

Contour Representations and Reduction Formulas

To evaluate these sums, the source utilizes contour integrals of the form: Z(m|t) = ((-1)^m-1 / 2πi) ∮ [ d log Ξ(1/2 + (t² + w)^1/2) / w^m ]. This representation allows for the computation of superzeta values without knowing individual zero locations. Furthermore, the paper provides reduction formulas that express the spectral sums of the second kind (𝒵) as finite sums of the first kind (Z), bridging the gap between the vertical distribution of zeros and their distance from the critical line.

Novel Research Pathways

1. Generalized Keiper-Li Positivity Scans

Formulation: Define a generalized sequence λ_n(t) based on the Taylor expansion of log Ξ at varying points 1/2 + t. RH is equivalent to λ_n(1/2) ≥ 0 for all n.

Methodology: Use the superzeta Z(s|t) to analyze the stability of these coefficients as t approaches 0. A "sign-flip" in λ_n(t) for small t would provide a definitive signature of a zero located off the critical line.

2. Spectral Determinant and Operator Construction

Formulation: The identity Z'(0|t) = -(1/2)log(2π)t + (1/4)log(8π) - log Ξ(1/2 + t) suggests Ξ is a determinant of a specific operator.

Methodology: Investigate the potential for a self-adjoint operator whose spectral zeta function matches Z(s|t). Proving the self-adjointness of such an operator would verify that all τ_k are real, thereby proving RH.

3. Rigidity through Analytic Continuation

Formulation: Quantify the "off-line defect" by comparing the meromorphic continuation of Z(s|t) with its spectral counterpart 𝒵(σ|t).

Methodology: If RH holds, the two functions must satisfy a strict set of reduction identities. Any deviation from these identities in the complex plane would indicate the existence of zeros with σ not equal to 1/2.

Computational Implementation

The following Wolfram Language code computes the superzeta value Z(2|t) using both the truncated sum of Riemann zeros and the transcendental derivative formula from arXiv:cea-01076225v1_document_document.

(* Section: Superzeta Special Value Comparison *)
(* Purpose: Validate the transcendental formula for Z(2|t) *)

ClearAll[xi, superZetaFormula, zeroSum];

(* Completed Zeta Function Xi[s] *)
xi[s_] := 1/2 * s * (s - 1) * Pi^(-s/2) * Gamma[s/2] * Zeta[s];

(* Superzeta Formula: Z(2|t) = - (d^2/ds^2) Log[Xi[s]] at s=1/2+t *)
superZetaFormula[t_] := Module[{s},
  - Derivative[2][Log[xi[#]] &][1/2 + t]
];

(* Truncated sum over first 500 non-trivial zeros *)
zeroSum[t_, num_] := Module[{zeros, s0},
  s0 = 1/2 + t;
  zeros = ZetaZero[Range[num]];
  (* Include both rho and its conjugate 1-rho *)
  Total[Table[(s0 - r)^(-2) + (s0 - (1 - r))^(-2), {r, zeros}]]
];

(* Execution and Comparison *)
tVal = 0.5;
valFormula = N[superZetaFormula[tVal], 10];
valSum = N[zeroSum[tVal, 500], 10];

Print["Formula Value: ", valFormula];
Print["Truncated Sum Value: ", valSum];
Print["Relative Error: ", Abs[(valFormula - valSum)/valFormula]];

(* Plotting the behavior of Z(2|t) *)
Plot[superZetaFormula[t], {t, 0.1, 1.5}, 
  PlotLabel -> "Superzeta Z(2|t) Landscape", 
  AxesLabel -> {"t", "Z(2|t)"}]

Conclusions

The study of superzeta functions as presented in arXiv:cea-01076225v1_document_document provides a powerful and unified framework for analyzing the Riemann zeros as a collective ensemble. By establishing the rational-transcendental duality of special values, the paper demonstrates that the zeros are not merely isolated points but are deeply embedded in the analytic structure of the completed zeta function. The most promising avenue for future research is the generalized Keiper-Li criterion, which utilizes the parameter t to scan for off-line zeros. Future work should focus on rigorous bounds for the finite-part evaluations at s=1, as these terms carry the most significant information regarding the local density of zeros near the critical line.

References

• arXiv:cea-01076225v1_document_document - Superzeta Functions and Special Values Analysis
• Keiper, J. B. (1992). "Power series expansions of Riemann's xi function." Mathematics of Computation.
• Li, X.-J. (1997). "The positivity of a sequence of numbers and the Riemann Hypothesis." Journal of Number Theory.
• Deninger, C. (1992). "Local L-factors of motives and regularized determinants." Inventiones Mathematicae.