Beyond the Critical Line: Asymptotic Convergence and the Vanishing of Zeta Zeros

Introduction

The Riemann Hypothesis remains one of the most profound challenges in mathematics, asserting that the non-trivial zeros of the Riemann zeta function zeta(s) lie on the critical line where the real part of s is 1/2. The research paper arXiv:hal-00476252v3 introduces a distinctive analytical framework that connects the distribution of these zeros to the asymptotic properties of the Hurwitz zeta function zeta(s, x) as the parameter x approaches 1.

The core motivation of this study is the realization that the behavior of the zeta function near its non-trivial zeros can be characterized by the rate of convergence of certain power series expansions as they approach the boundary of their convergence disk. By establishing a remarkable identity that relates the limit of the ratio of the zeta function to a specially constructed auxiliary function, the author attempts to demonstrate that a violation of the Riemann Hypothesis would lead to a logical contradiction in the growth rates of these functions.

Mathematical Background

The foundation of this study involves the Hurwitz zeta function, defined as the sum of (n+x) to the power of negative s. The paper utilizes an integral representation of the zeta function involving a kernel function related to the generating function of the Bernoulli numbers. A key sequence in this analysis is S_n(s), an alternating sum of (k+1) to the power of negative s, which admits a representation as an integral of (1 - e to the power of negative t) raised to the power of n-1.

The relationship between these finite difference sums and the Riemann zeta function is given by a critical identity: the infinite sum of S_n(s)/(n+1) is equal to (s-1) multiplied by zeta(s). This identity is crucial because it connects the global property of the zeta function, specifically its zeros, to the convergence of a series whose terms are defined by finite differences.

Main Technical Analysis

Spectral Properties and the Remarkable Identity

The core technical result is a limit identity involving a gauge function Phi(x). For a non-trivial zero s, the limit as x approaches 1 of (s-1)zeta(s, x)/Phi(x) is shown to be -1/(s Gamma(s)). This identity suggests that Phi(x) acts as a measure of the vanishing rate of the zeta function near its zeros. By comparing the convergence rates of symmetric zeros s and 1-s, the analysis aims to show that any discrepancy in the real part of s leads to a contradiction in the growth rates of these functions.

Moment Estimates and Coefficient Asymptotics

The analysis further derives coefficients a_n defined through the n-th derivative of an integral involving the kernel function. By evaluating the asymptotic behavior of these coefficients for large n, the paper shows they are intrinsically linked to the values of the Riemann zeta function at even integers via Bernoulli numbers. This provides a controlled decay rate essential for ensuring the convergence of the series representations used to investigate the critical strip.

Novel Research Pathways

Pathway 1: Bernoulli-Zeta Correspondence Theory

The connection between Bernoulli number asymptotics and zeta function zeros suggests a correspondence where deviations in coefficient growth rates encode zero distribution information. Future research could involve computing high-precision values of these coefficients to analyze their spectral properties. If a zero exists off the critical line, it should manifest as a specific oscillatory frequency in the deviation sequence of these coefficients.

Pathway 2: Geometric Convergence Measures

The auxiliary function Phi(x) defines a geometric correspondence between points in the critical strip and measures on the unit interval. Analyzing the Riemannian geometry of this manifold could reveal why the critical line represents a unique phase boundary. The hypothesis is that zeros on the critical line correspond to optimal geometric configurations where the convergence measure achieves minimal curvature.

Computational Implementation

(* Section: Zeta Zero Convergence Analysis *)
(* Purpose: Verify Sn(s) series and visualize the Phi(x) limit *)

ClearAll[Sn, ZetaSeries, Phi, testS, x];

(* Define the finite sum Sn(s) as per arXiv:hal-00476252v3 *)
Sn[n_Integer, s_] := Sum[(-1)^k * Binomial[n - 1, k] * (k + 1)^(-s), {k, 0, n - 1}];

(* Function to compute the partial sum of the Sn series *)
ZetaSeries[limit_Integer, s_] := Sum[Sn[n, s]/(n + 1), {n, 1, limit}];

(* Parameters for testing: s on the critical line (first zero) *)
testS = 1/2 + 14.134725141734693*I;

(* Compare partial sum to (s-1)zeta(s) *)
targetValue = (testS - 1) * Zeta[testS];
Print["Target Value (s-1)Zeta(s) at first zero: ", NumberForm[targetValue, 5]];
Print["Partial Sum (n=50): ", NumberForm[ZetaSeries[50, testS], 5]];

(* Visualizing the Lerch Zeta limit behavior as x approaches 1 *)
Phi[x_, s_] := (1 - x) * (Log[1 - x] / -x)^(1 - s);

Plot[Abs[(testS - 1) * LerchPhi[x, testS, 1] / Phi[x, testS]], {x, 0.9, 0.999}, 
  PlotRange -> All, 
  Frame -> True, 
  PlotLabel -> "Limit Behavior of Lerch Zeta Ratio near x=1"]

Conclusions

The framework presented in arXiv:hal-00476252v3 offers a compelling new lens through which to view the Riemann Hypothesis. By focusing on the convergence dynamics of the Hurwitz zeta function, it shifts the problem from global contour integration to local asymptotic analysis. The most promising next step is the refinement of the error bounds for the limit identity to rigorously evaluate the symmetry between zeros s and 1-s. Ultimately, the transition from the parameter x to the limit of 1 offers a unique microscopic view of the zeta function's behavior, potentially providing the tools necessary to finally resolve the location of its non-trivial zeros.

References

• arXiv:hal-00476252v3
• Edwards, H.M. (1974). Riemann's Zeta Function. Academic Press.
• Titchmarsh, E.C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.