Scaling the Critical Line: Polylogarithmic Asymptotics and the Riemann Hypothesis

Introduction

The study of the Riemann zeta function, ζ(s), has historically oscillated between global analytic perspectives provided by the functional equation and the local arithmetic properties of Dirichlet series. The research presented in arXiv:hal-01355277v1 bridges these domains by introducing a parametrization of the zeta function using the polylogarithm function Li_s(z), specifically focused on the transformation of the argument z/(z-1). This approach allows for an investigation of the non-trivial zeros within the critical strip by examining the asymptotic behavior of a parameterized function Z(s, x) as the parameter x approaches the singular point at 1.

By mapping the discrete properties of the zeta function onto the continuous analytic landscape of polylogarithms, this framework provides a novel criterion for the critical line σ = 1/2. The primary motivation is to identify a local indicator that distinguishes the critical line from other regions within the critical strip 0 < Re(s) < 1, leveraging the symmetry inherent in the functional equation through a limit process.

Mathematical Background

The foundation of this analysis rests on the polylogarithm function, Li_s(z), defined for |z| < 1 by the series ∑ zⁿ/n^s. The paper arXiv:hal-01355277v1 utilizes a specific transformation, Li_s(z/(z-1)), which maps the unit disk in a manner that facilitates the study of the zeta function's analytic continuation.

Key Series Representations

The paper highlights several series for the zeta function that converge globally. A fundamental structure is the sum:

• S_n(s): Defined as the alternating sum ∑ (-1)^k (n-1 choose k) (k+1)^-s.
• Hasse Representation: (s-1)ζ(s) = ∑ (1/(n+1)) S_n+1(s-1).
• The Z(s, x) Function: An Abel-parameterized version Z(s, x) = ∑ (S_n(s)/(n+1)) xⁿ⁺¹.

As x approaches 1, Z(s, x) tends toward (s-1)ζ(s). However, the specific path of this limit is sensitive to the real part of s, creating a mechanism to probe the location of non-trivial zeros.

Main Technical Analysis: The Log-Power Trichotomy

The core technical contribution of arXiv:hal-01355277v1 is the comparison of Z(s, x) with its dual Z(1-s, x). This comparison effectively tests the functional equation under a limit process as x approaches the boundary at 1.

Singularities at the Boundary

For values of s in the critical strip, the function Li_s(z/(z-1)) exhibits a singularity as z approaches 1. The analysis establishes that the leading behavior is governed by the relation:

Li_s(z/(z-1)) ≈ -(-ln(1/z - 1))^s / Γ(s+1)

Applying this to the parameterized function Z(s, x), we find that Z(s, x) is composed of a principal part involving the zeta function and a singular part involving the logarithmic growth. When ζ(s) vanishes at a non-trivial zero s₀, the singular part becomes the dominant term in the expansion.

The Ratio Test for the Critical Line

Consider a potential non-trivial zero s₀ = σ₀ + i t₀. At this point, the ratio of the parameterized functions |Z(s₀, x) / Z(1-s₀, x)| scales according to the exponent 2σ₀ - 1. This leads to a sharp trichotomy:

• If σ₀ < 1/2: The ratio tends to 0 as x approaches 1.
• If σ₀ > 1/2: The ratio tends to ∞ as x approaches 1.
• If σ₀ = 1/2: The ratio tends to a finite, non-zero constant.

This result implies that the symmetry of the functional equation is only preserved in the Abel-summed limit if the zero lies exactly on the critical line. For any zero off the line, the parameterized approximation breaks the symmetry as it approaches the boundary point.

Novel Research Pathways

The methodology introduced in arXiv:hal-01355277v1 suggests several concrete directions for future investigation into the Riemann Hypothesis.

Pathway 1: Spectral Analysis of Integrated Polylogarithms

One promising avenue is to develop a spectral theory for the integral operator associated with the polylogarithmic kernel. By studying the eigenvalues of an operator defined by the integral of Li_s(z/(z-1)), researchers might identify a trace formula that connects the operator's spectrum directly to the zeros of the zeta function. This would provide a Hilbert-Polya style approach grounded in polylogarithmic theory.

Pathway 2: Uniformity and Error Bounds

A rigorous proof of RH using this method would require establishing uniform error bounds for the asymptotic expansion of Z(s, x) across the entire critical strip. Current work relies on leading-order terms; refining these bounds to account for higher-order logarithmic corrections could eliminate the possibility of off-line zeros satisfying the boundary conditions.

Computational Implementation

The following Wolfram Language code demonstrates the Abel parametrization and the ratio test. It approximates Z(s, x) and visualizes how the ratio |Z(s, x)/Z(1-s, x)| behaves for values on and off the critical line.

(* Section: Polylogarithmic Zeta Analysis *)
(* Purpose: Compute Z(s,x) and verify critical line behavior *)

SeriesS[n_, s_] := Sum[Binomial[n-1, m-1] (-1)^(m-1) m^(-s), {m, 1, n}];

ZetaZ[s_, x_, nmax_:60] := Module[{terms, result},
  terms = Table[SeriesS[n, s]/(n+1), {n, 1, nmax}];
  result = (s-1) Zeta[s] + (x-1) Sum[terms[[n]] * x^n, {n, 1, nmax}];
  result
];

CriticalLineTest[t_, x_] := Module[{s1, z1, z2, ratio},
  s1 = 1/2 + I t;
  z1 = ZetaZ[s1, x];
  z2 = ZetaZ[1 - s1, x];
  Abs[z1/z2]
];

(* Compare a zero on the line vs a point off the line *)
Module[{t1 = 14.134725, xvals, onLine, offLine},
  xvals = Range[0.8, 0.98, 0.02];
  onLine = Table[CriticalLineTest[t1, x], {x, xvals}];
  offLine = Table[Abs[ZetaZ[0.7 + I t1, x] / ZetaZ[0.3 - I t1, x]], {x, xvals}];
  
  Print["Ratio values as x -> 1:"];
  Print["On Critical Line (σ=0.5): ", onLine];
  Print["Off Critical Line (σ=0.7): ", offLine];
]

Conclusions

The analysis of arXiv:hal-01355277v1 reframes the Riemann Hypothesis as a boundary stability problem. By utilizing polylogarithmic series, the research establishes that the critical line is the unique locus where the singular components of the zeta function's Abel parametrization remain balanced. While challenges regarding uniform convergence and limit interchanges remain, the log-power trichotomy provides a precise analytic mechanism to distinguish the critical line. Future research focusing on higher-order asymptotics and spectral properties of the Z(s, x) function may offer the rigorous path forward needed to resolve the hypothesis.

References

• arXiv:hal-01355277v1: Polylogarithmic series representations and Riemann zeta function analysis.
• Hasse, H. (1930). Ein Summierungsverfahren für die Riemannsche ζ-Reihe. Mathematische Zeitschrift.
• Lewin, L. (1981). Polylogarithms and Associated Functions. North-Holland.