Combinatorial Symmetries and the Critical Line: Linking Unbordered Words to the Riemann Hypothesis

Introduction

The study of formal languages and combinatorics on words has traditionally been viewed as a discipline distinct from analytic number theory. However, recent developments in the analysis of monic irreducible polynomials over finite fields have revealed deep underlying connections between these fields. The research presented in arXiv:hal-00990604v1, titled Unbordered words and monic irreducible polynomials, provides a rigorous framework for understanding the density of unbordered words and their structural isomorphism to irreducible polynomials.

At the heart of this investigation is the distribution of these discrete objects as the length of the word (or the degree of the polynomial) tends toward infinity. In number theory, the distribution of prime numbers is governed by the Riemann Hypothesis (RH), which posits that the non-trivial zeros of the Riemann zeta function lie on the critical line with real part 1/2. In the context of function fields, the analogous Riemann Hypothesis for curves over finite fields provides an exact bound on the error terms for the Prime Polynomial Theorem, mirroring the patterns found in word combinatorics.

Mathematical Background

To bridge the gap between the source paper and the Riemann Hypothesis, we must first define the primary objects of study. Let Sigma be an alphabet of size k. A word w is said to be bordered if there exists a non-empty word u such that u is both a proper prefix and a proper suffix of w. Conversely, an unbordered word is one where no such u exists.

The paper arXiv:hal-00990604v1 explores the set D_n of unbordered words of length n. The counting function for these words is related to the number of monic irreducible polynomials of degree n over a finite field F_q. The Prime Polynomial Theorem states that the number of such polynomials is approximately q^n/n. The Riemann Hypothesis for function fields manifests in the observation that the subsequent error terms are bounded by q^(n/2)/n, a square-root suppression that is a hallmark of the critical line.

Main Technical Analysis

The Square-Root Law in Word Densities

One of the most striking inequalities provided in the source paper is the bound for the sum of powers related to the divisors of the word length i:

2((k^((i/2)+1) - 1) / (k - 1)) + sum over d|i (i * k^(i/d)) is less than or equal to (2k / (k - 1)) * k^(i/2) + i^2 * k^(i/3)

This inequality is mathematically significant because it isolates the k^(i/2) term as the dominant error term. In the context of the Riemann Hypothesis, the exponent 1/2 represents the critical line. If we consider the alphabet size k as the base of the field q, the term k^(i/2) represents the square-root of the total space of words, indicating that the fluctuations in word patterns are governed by the same spectral constraints as prime number distribution.

Theorem 8 and the Function-Field Riemann Hypothesis

The paper leverages Theorem 8, which counts monic irreducible polynomials of degree N congruent to a modulo m. This is a function-field version of Dirichlet's Theorem on Primes in Arithmetic Progressions. The proof of such counting results relies on L-functions where the reciprocal zeros satisfy an exact analogue of RH: they lie on a circle with radius q^(-1/2). The fact that the paper finds |D_n| = Omega((k - epsilon)^(n/2)) suggests that the fluctuations in unbordered words are large enough to account for the randomness required by RH, yet remain strictly bounded by the square-root law.

Novel Research Pathways

Pathway 1: The Border Zeta Function

We propose the definition of a Border Zeta Function, an exponential generating object that encodes the counts of unbordered words. If borderlessness can be captured by a finite-state presentation, this zeta function will be meromorphic with singularities governed by an operator spectrum. An RH-type statement in this context would assert that all nontrivial singularities lie on the circle |z| = k^(-1/2), providing a combinatorial proof of zero distribution in a new setting.

Pathway 2: Spectral Gaps and Prime Gaps

Unbordered words can be viewed as specific paths in a De Bruijn graph that avoid certain loop-back conditions. Investigating the spectral gap of the transition matrix for these paths may yield new insights into the Montgomery Pair Correlation Conjecture. If the spectral gap is exactly 1/2, it would establish a direct link between the repulsion of zeta zeros and the non-overlap properties of combinatorial strings.

Computational Implementation

The following Wolfram Language code demonstrates the relationship between the density of irreducible polynomials (which satisfy the RH) and the square-root error bounds discussed in arXiv:hal-00990604v1.

(* Section: Irreducible Polynomial Density and Riemann Bounds *)
(* Purpose: Compare N_q(n) error terms to the q^(n/2) bound *)

Module[{q = 2, maxDegree = 20, irreducibles, primePolynomialTheorem, errorTerm, rhBound},
  
  (* Calculate N_q(n) using Mobius inversion *)
  irreducibles = Table[
    (1/n) * Total[MoebiusMu[n/Divisors[n]] * q^Divisors[n]], 
    {n, 1, maxDegree}
  ];
  
  (* Leading term: q^n/n *)
  primePolynomialTheorem = Table[q^n/n, {n, 1, maxDegree}];
  
  (* Observed error vs RH bound q^(n/2)/n *)
  errorTerm = Abs[irreducibles - primePolynomialTheorem];
  rhBound = Table[(q^(n/2))/n, {n, 1, maxDegree}];
  
  (* Plot results on log scale *)
  ListLogPlot[
    {irreducibles, primePolynomialTheorem, errorTerm, rhBound},
    PlotLegends -> {"Actual Irreducibles", "Leading Term", "Observed Error", "RH Bound"},
    Joined -> True, 
    Frame -> True, 
    PlotLabel -> "Polynomial Density vs. Riemann Hypothesis Bounds"
  ]
]

Conclusions

The analysis of arXiv:hal-00990604v1 reveals that the combinatorial structure of unbordered words is a fundamental reflection of the arithmetic properties of finite fields. The paper's derivation of the k^(n/2) error term provides a concrete link to the Riemann Hypothesis for function fields. By demonstrating that the border-free property in formal languages mirrors the irreducibility property in algebra, we open a path for using combinatorial tools to attack problems in analytic number theory. The most promising next step is the spectral analysis of word-transition matrices to prove the non-vanishing of zeta functions in complex domains.

References

• Source Paper: arXiv:hal-00990604v1
• M. Rosen, Number Theory in Function Fields, Graduate Texts in Mathematics, Springer, 2002.
• A. Weil, Sur les courbes algebriques et les varietes qui s'en deduisent, 1948.