Spectral Determinants and Generalized Zeta Functions: A New Framework for the Riemann Hypothesis

Executive Summary

The research article based on arXiv:1403.4558 presents a profound synthesis of spectral theory and analytic number theory by constructing generalized zeta functions defined over the nontrivial zeros of the Riemann zeta function. The central insight is the definition of the structure Z(s | t) = sum (1/2 + t - rho)^(-s), which acts as a bridge between the discrete distribution of zeros and the global analytic properties of the Riemann Xi function. This approach is particularly promising because it unifies historical approaches—such as the Keiper-Li sequence and Deninger program—into a coherent spectral determinant framework. By regularizing these divergent sums, the source paper provides a machinery to map the Riemann Hypothesis (RH) onto the positivity and monotonicity of specific analytic continuations, offering a rigorous pathway to investigate the critical line through the lens of functional analysis and spectral geometry.

Introduction

The Riemann Hypothesis remains the most significant unsolved problem in mathematics, asserting that all nontrivial zeros of the Riemann zeta function zeta(s) lie on the critical line with real part 1/2. Traditionally, the study of these zeros, denoted as rho, has been approached through prime number density and explicit formulas. However, a parallel tradition treats these zeros as the spectrum of a hypothetical physical or mathematical operator. The paper arXiv:1403.4558, authored by Andre Voros, formalizes this spectral perspective by developing the theory of "zeta functions over zeros."

This article analyzes how these secondary zeta functions encode the distribution of zeros into the derivatives of the completed zeta function, Xi(s). By treating the zeros as spectral data, the paper constructs a dictionary between classical special functions—like the Hurwitz zeta function and Gamma function—and the structures arising from the Riemann zeros. This contribution is vital because it transforms the point-wise problem of locating individual zeros into a study of the global stability and growth properties of a parameterized family of functions. We will examine how this framework facilitates new research pathways into the Keiper-Li criterion and the spectral determinant interpretation of the Riemann Xi function.

Mathematical Background

To understand the technical depth of arXiv:1403.4558, we must first define the primary mathematical objects. The Riemann Xi function is an entire function defined to eliminate the poles and trivial zeros of the zeta function: Xi(s) = (1/2) s (s - 1) pi^(-s/2) Gamma(s/2) zeta(s). The zeros of Xi(s) are precisely the nontrivial zeros rho of zeta(s), and they satisfy the functional equation Xi(s) = Xi(1 - s).

The core construction in the paper is the generalized secondary zeta function, Z(s | t), defined as the sum over all nontrivial zeros rho of the terms (1/2 + t - rho)^(-s). Here, t is a complex parameter that shifts the origin of the spectral sum. When t = 1/2, we obtain the specialized sum Z_*(s) over (1 - rho)^(-s). The paper demonstrates that while these sums are initially divergent for small Re(s), they can be meromorphically continued to the entire complex s-plane.

The paper establishes a striking analogy between Z(s | t) and the Hurwitz zeta function zeta(s, w). Just as the Hurwitz zeta function at negative integers yields Bernoulli polynomials, Z(s | t) at negative integers yields regularized values that depend on the Taylor coefficients of log Xi(s). This parallel suggests that the set of Riemann zeros behaves like a structured sequence of eigenvalues, allowing the application of zeta-regularization techniques common in quantum field theory and spectral geometry.

Main Technical Analysis

Spectral Properties and Zero Distribution

The distribution of the zeros rho is captured by the values of Z(s | t) at positive integers. One of the most operational identities derived in arXiv:1403.4558 relates these spectral sums to the logarithmic derivatives of the Xi function. Specifically, for an integer m > 1, the value Z(m | t) is proportional to the m-th derivative of log Xi evaluated at 1/2 + t. Formally: Z(m | t) = ((-1)^(m-1) / (m-1)!) (d^m/dz^m) log Xi(z) at z = 1/2 + t.

This identity implies that the local clustering of zeros near any point on the critical line is encoded in the local growth rate of the Xi function. If the Riemann Hypothesis holds, all zeros rho take the form 1/2 + i*tau, making the terms in the sum (t - i*tau)^(-s). This configuration imposes strict constraints on the analytic behavior of Z(s | t), particularly ensuring that certain derivatives maintain a specific sign, which is a hallmark of spectral stability.

The Keiper-Li Sequence and Generating Functions

The Keiper-Li sequence, lambda_n, is a set of numbers whose positivity for all n is equivalent to the Riemann Hypothesis. The paper arXiv:1403.4558 shows that these coefficients are embedded within the generating function of the secondary zeta function. By expanding log Xi(1 / (1-z)), the author establishes that lambda_n can be expressed as a finite combination of the values of Z_*(m) for m up to n.

This connection is crucial because it moves the Li criterion from a discrete sequence of inequalities to a continuous condition on the analytic continuation of Z(s | t). The "Finite Part" (FP) of Z(s | t) at the pole s = 1 acts as a primary regulator for the growth of lambda_n. The paper provides explicit tables showing how Z(s | t) values at negative integers (regularized) and derivatives at zero (transcendental) recover the fundamental constants of the zeta function, such as Euler's constant and log pi.

Evenized Spectral Data and Determinants

Another technical breakthrough in the source is the "evenization" of the spectral data. The paper defines Z_even(sigma | t) as the sum over (tau_k^2 + t^2)^(-sigma), where tau_k are the imaginary parts of the zeros. This structure aligns the Riemann zeros with the theory of spectral determinants. In this context, the Riemann Xi function can be viewed as the determinant of an operator whose eigenvalues are tau_k^2. Voros proves that the derivative of Z_even at sigma = 0 recovers log Xi(1/2 + t), effectively representing the Xi function as a zeta-regularized product over its zeros. This provides a rigorous foundation for the Hilbert-Polya conjecture, suggesting that the zeros are indeed the spectrum of a self-adjoint operator.

Novel Research Pathways

Pathway 1: Parameter Modulation and Stability Criteria

A promising research direction involves studying the Stability Transition as the parameter t in Z(s | t) approaches the critical value of 0. Since t = 0 corresponds exactly to the critical line, one can formulate a stability criterion: the Riemann Hypothesis is true if and only if the function f(t) = Z(1 | t) remains holomorphic and monotonic for all Re(t) > 0. This would allow researchers to use techniques from the theory of dynamical systems and stability analysis to detect potential off-line zeros as "singularities" in the t-parameter space.

Pathway 2: Complete Monotonicity of Evenized Transforms

Building on the evenized spectral zeta function Z_even(sigma | t), a new pathway involves proving the Complete Monotonicity of the log-Xi derivatives. If RH holds, the sequence of squared imaginary parts is positive. For such positive spectra, the resulting zeta function often exhibits complete monotonicity (alternating signs in derivatives). Establishing this property for Z_even would provide a new family of RH-equivalent inequalities that could be tested using asymptotic expansions and high-precision numerical methods.

Pathway 3: Arithmetic of Zero-Bernoulli Polynomials

The source paper demonstrates that Z(-n | t) yields polynomials in t, which we may call Zero-Bernoulli polynomials. Just as classical Bernoulli numbers characterize the values of zeta(s) at integers, these polynomials characterize the moments of the zero distribution. Investigating the recurrence relations and algebraic properties of these polynomials could lead to a recursive method for calculating Keiper-Li coefficients without needing to locate individual zeros, potentially leading to a purely algebraic proof of their positivity.

Computational Implementation

The following Wolfram Language code implements the calculation of the generalized secondary zeta function Z(m | t) by relating it to the derivatives of the Riemann Xi function, as established in arXiv:1403.4558.

(* Section: Secondary Zeta Function and Xi Derivatives *)
(* Purpose: Demonstrate the identity Z(m | t) proportional to Log[Xi] derivatives *)

(* Define the completed Riemann Xi function *)
XiFunction[s_] := 1/2 * s * (s - 1) * Pi^(-s/2) * Gamma[s/2] * Zeta[s];

(* Define the generalized secondary zeta function Z(m | t) *)
(* Using the identity: Z(m | t) = ((-1)^(m-1)/(m-1)!) * d^m/ds^m Log[Xi[s]] *)
SecondaryZeta[m_Integer, t_?NumericQ] := Module[{s, logXi, val},
  logXi = Log[XiFunction[s]];
  val = ((-1)^(m - 1) / Factorial[m - 1]) * (Derivative[m][Log[XiFunction[#]] &][1/2 + t]);
  N[val, 20]
];

(* Compute a table of Z(m | t) for m=2 at various spectral shifts t *)
tValues = {0.5, 1.0, 1.5, 2.0};
zTable = Table[{t, SecondaryZeta[2, t]}, {t, tValues}];

Print["Table of Z(2 | t) values:"];
Print[TableForm[zTable, TableHeadings -> {"t", "Z(2 | t)"}]];

(* Plot the secondary zeta function Z(1 | t) to observe its decay *)
Plot[Re[SecondaryZeta[1, t]], {t, 0.1, 5}, 
  PlotLabel -> "Secondary Zeta Function Z(1 | t)", 
  AxesLabel -> {"t", "Z(1 | t)"}, 
  PlotStyle -> {Thick, Blue}]

Conclusions

The analysis of generalized zeta functions over Riemann zeros provided by arXiv:1403.4558 offers a robust and elegant framework for tackling the Riemann Hypothesis. By unifying the discrete data of the zeros into a continuous analytic structure, the paper allows for the application of spectral determinant theory and regularization methods. The most promising avenue for future research lies in the study of the evenized spectral zeta functions and their monotonicity properties, which directly link the critical line to the stability of the Xi function.

Ultimately, the secondary zeta function Z(s | t) serves as a powerful diagnostic tool. It transforms the search for zeros into the study of a well-behaved analytic function whose properties are governed by the global symmetry of the zeta function itself. Further exploration of the Zero-Bernoulli polynomials and the t-parameter stability may finally provide the constraints necessary to confirm the distribution of zeros on the critical line.

References

• arXiv:1403.4558: Voros, A. "Zeta functions over the Riemann zeros."
• Keiper, J. B. (1992). "Power series expansions of Riemann's xi function." Journal of Number Theory.
• 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.