Arithmetic Heights and Beta-Conjugate Spacing: A Dynamical Approach to Zeta Zeros

Introduction

The study of β-expansions, originally initiated by Renyi and Parry, has transitioned from a niche topic in symbolic dynamics to a powerful framework for understanding the distribution of algebraic integers. The research paper arXiv:hal-00341880 provides a rigorous exploration of beta-conjugates—the roots of Parry polynomials—and their clustering behavior. This analysis establishes a bridge between discrete dynamical systems and analytic number theory, specifically the distribution of zeros of the Riemann zeta function ζ(s).

The central problem addressed in the source is the characterization of the height and factorization of the Parry polynomial n_β^*(z). This polynomial encapsulates the arithmetic essence of the base β. By analyzing the distribution of its roots, specifically those lying within or near the unit circle, the paper establishes bounds that resonate with classical problems such as the Riemann Hypothesis (RH). This synthesis proposes that the spectral properties of Parry polynomials provide a pathway to understanding the vertical distribution of zeta zeros through the lens of Mahler measures and Erdos-Turan discrepancy.

Mathematical Background

To connect arXiv:hal-00341880 to the Riemann Hypothesis, we define the key algebraic objects. A Parry number β > 1 is a real number such that the expansion of 1, denoted d_β(1), is eventually periodic. This expansion is represented as a sequence t₁, t₂, t₃, ... where the coefficients are determined by the greedy map.

The Parry polynomial n_β^*(z) is the monic polynomial whose roots are the beta-conjugates of β. As detailed in the source, this polynomial can be factored into cyclotomic polynomials Φ_n(z), reciprocal factors κ_j^*(z), and non-reciprocal factors g_j(z). A critical result is that the height H(n_β^*) is constrained to the set {floor(β), ceil(β)}. This implies that the coefficients are remarkably small relative to the degree, mirroring the behavior of the Taylor coefficients of the Riemann ξ-function.

The Mahler measure M(P) of these polynomials is related to Jensen's Formula, which states that the mean of log |P| on the unit circle equals the sum of the logs of the moduli of the roots outside that circle. In the context of RH, this measure serves as a proxy for the growth rate of the zeta function along the critical line Re(s) = 1/2.

Main Technical Analysis

Spectral Properties and Zero Distribution

The distribution of beta-conjugates in arXiv:hal-00341880 exhibits a clustering phenomenon near the unit circle. This is formalized through the study of the Rauzy fractal and the central tile Ω'. The paper provides a geometric bound where the degree d_P is limited by the number of lattice points in a projected region of this tile. This is fundamentally a spectral result; the central tile is the state space of a dynamical system whose spectrum is determined by the beta-conjugates.

The source also introduces a bound for the height h(G) involving the Möbius function μ(j). Specifically, the bound involves the term sum μ(j)/j², which is exactly 1/ζ(2). This suggests that the density of the Parry polynomial's coefficients is governed by the same square-free distribution found in the prime number theorem. Since the RH is equivalent to specific error terms in prime distributions, the fluctuations in these heights may provide a discrete model for testing the fluctuations of the Mertens function.

Multiplicity Constraints and Height Bounds

A distinctive feature of the source paper is the quantitative constraint on the multiplicity ν_ξ of roots. For beta-conjugates that are not units, the multiplicity is bounded by:
ν_ξ ≤ (1/log 2) * (log H(n_β^*) - log |N(β)|).

This bound becomes particularly restrictive when the norm of β is large, forcing non-unit conjugates to be simple roots. This parallels the Simplicity Conjecture for the zeros of the Riemann zeta function. Furthermore, the paper demonstrates that the number of distinct cyclotomic factors s is bounded by an absolute constant times the square root of the degree d_P. This square-root law mirrors the expected cancellation in sums over zeta zeros, suggesting that the cyclotomic structure provides a discrete analog for the cancellation phenomena central to the RH.

Novel Research Pathways

1. Dynamical L-Functions from Beta-Conjugates

A promising research direction involves constructing L-functions directly from the distribution of beta-conjugates. By defining L_β(s) as a product over conjugates weighted by their multiplicities, one can create a function whose analytic properties mirror the Riemann zeta function. The height bounds from arXiv:hal-00341880 provide the growth estimates necessary to control these functions in vertical strips.

2. Spectral Gaps in Rauzy Fractals

The topology of the Rauzy fractal central tile—specifically whether it contains holes—is linked to the distribution of conjugates. Investigating the spectral gap of the transfer operator associated with the β-shift could provide a new way to define zero-free regions. A "solid" fractal with no holes would correspond to a maximal spectral gap, potentially mirroring the conditions required for the RH to hold.

3. Möbius Cancellation and the Mertens Criterion

The tilde-h bound in the source involves nested sums of Möbius weights. One could replace the absolute values in these bounds with conditional estimates assuming the Riemann Hypothesis. This would allow researchers to deduce if RH-level cancellation in the Möbius function leads to tighter bounds on the angular equidistribution of beta-conjugates, establishing a two-way street between dynamical systems and prime number theory.

Computational Implementation

(* Section: Parry Polynomial Roots and Zeta Spacing Comparison *)
(* Purpose: This code computes roots of Parry-type polynomials *)
(* and compares their distribution to Riemann Zeta zeros. *)

Module[{
  delta = 5, 
  poly, 
  roots, 
  zetaZeros, 
  spacings, 
  normSpacings
},
  (* Define a Selmer-type Parry polynomial from the paper *)
  poly = x^(delta + 2) - x^(delta + 1) - 1;
  
  (* Solve for roots numerically *)
  roots = x /. NSolve[poly == 0, x];
  
  (* Extract imaginary parts of first 50 Zeta zeros *)
  zetaZeros = Table[Im[ZetaZero[n]], {n, 1, 50}];
  
  (* Calculate normalized spacings of Zeta zeros *)
  spacings = Differences[zetaZeros];
  normSpacings = Table[
    spacings[[i]] * (Log[zetaZeros[[i]] / (2 * Pi)] / (2 * Pi)), 
    {i, Length[spacings]}
  ];

  (* Output the results *)
  Print["Parry Polynomial Roots: ", roots];
  Print["Mean Normalized Zeta Spacing: ", Mean[normSpacings]];

  (* Visualization of root distribution *)
  Show[
    Graphics[{Gray, Circle[{0, 0}, 1]}],
    ComplexListPlot[roots, PlotStyle -> Red, 
      PlotLabel -> "Beta-Conjugates vs Unit Circle"]
  ]
]

Conclusions

The analysis of arXiv:hal-00341880 reveals that Parry numbers and their associated polynomials are deeply embedded in the landscape of analytic number theory. The factorization of the Parry polynomial into cyclotomic and non-reciprocal parts provides a structural parallel to the decomposition of the Riemann zeta function into trivial and non-trivial zeros.

The most promising avenue for further research lies in the arithmetic height estimates. The explicit inclusion of the Möbius function suggests that the Parry polynomial's height captures the same pseudo-randomness found in the distribution of prime numbers. Future work should focus on the "Parry-Zeta" limit, observing if conjugate distributions converge toward the statistical benchmarks expected of the Riemann zeta function.

References

• arXiv:hal-00341880: Conjugates of a Perron number and Erdõs-Turán approach.
• Boyd, D. W. (1981). Speculations on the distribution of the heights of Pisot numbers. Mathematics of Computation.
• Schinzel, A. (2000). Polynomials with Special Regard to Reducibility. Cambridge University Press.