The Automata of Primes: Multiplicative Rationality and the Riemann Critical Line

Introduction

The study of multiplicative structures within the natural numbers, denoted as N, represents a fundamental intersection between formal language theory and analytic number theory. The source paper arXiv:hal-00019898v1, titled "Rational sets of integers," establishes rigorous criteria for when the multiplicative closure of a set—specifically the right ideal LN—constitutes a "rational" set. In this context, rationality implies that the set can be expressed as a finite union of arithmetic progressions, a property that aligns with the capabilities of finite-state automata.

While the paper's primary focus is algebraic and combinatorial, the conditions it derives for the rationality of these sets have profound implications for the distribution of prime numbers. The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function, ζ(s), lie on the critical line Re(s) = 1/2. This assertion is fundamentally a statement about the regularity and density of primes. By examining the "prime power saturation" requirement identified in arXiv:hal-00019898v1, we can construct a bridge between the automata-theoretic view of integers and the spectral properties of the zeta function.

This article provides a comprehensive analysis of how the rationality of multiplicative ideals serves as a proxy for the analytic behavior of Dirichlet series. We demonstrate that the transition from a rational set to a non-rational set mirrors the introduction of complex oscillations in prime counting functions, offering a novel lens through which the critical line might be investigated.

Mathematical Background

To understand the connection to the Riemann Hypothesis, we must first define the core objects identified in arXiv:hal-00019898v1. Let N be the set of positive integers. A set S is considered rational, or Rat(N), if it is a finite union of sets of the form a + bN, where a is a non-negative integer and b is a positive integer. These are the sets recognizable by finite automata in a unary representation.

The source paper focuses on the set LN = { l * n | l in L, n in N }. This set represents the multiplicative closure of L. A central result of the paper is Theorem 6, which states that LN is an element of Rat(N) if and only if for every prime p less than or equal to m + q, there exists an exponent b ≥ 1 such that p^b is an element of L. Here, m and q define the initial structure of L as a union of a finite set I and arithmetic progressions P + qN.

This relates to the Riemann zeta function via the Euler product ζ(s) = Π (1 - p^-s)^-1. The Dirichlet series of the indicator function of a set S encodes its structural properties. If S is rational, its Dirichlet series is a finite linear combination of Hurwitz zeta functions, which are meromorphic with only a simple pole at s=1. However, if LN is not rational—specifically because it lacks certain prime powers—the associated Dirichlet series inherits the complex zero-distribution properties of the primes it excludes.

Main Technical Analysis

Spectral Properties and Dirichlet Analyticity

The core technical contribution of arXiv:hal-00019898v1 is the characterization of the set LN. Consider the Dirichlet series D_L(s) associated with the indicator function of LN. If LN is rational, then D_L(s) must be meromorphic in the entire complex plane. The paper shows that if L does not satisfy the "prime power saturation" condition, then LN is not rational. From an analytic perspective, this implies that D_L(s) cannot be simply represented as a finite sum of Hurwitz zeta functions.

This "non-rationality" introduces a higher level of complexity in the distribution of the elements of LN. The connection to the Riemann Hypothesis becomes explicit when we examine the density of these sets. For a rational set, the density is a rational number. If LN is not rational, its density might be determined by more complex analytic properties. The source paper's Theorem 11 states that (a + bN)N is rational if and only if a divides b or b divides a. This restrictive condition implies that the multiplicative action usually breaks the additive periodicity of arithmetic progressions, a phenomenon bounded by the distribution of ζ(s) zeros.

The gcd-Normalization and Local-Global Principles

In Section 3 of arXiv:hal-00019898v1, the authors discuss the structure of L_u, defined as I_u union (P_u + qN), where I_u = {x / gcd(u, x) | x in I}. This construction describes how the set L transforms under division by an integer u. This is essentially a study of operators acting on the space of sets of integers, analogous to Hecke operators in the theory of L-functions.

If LN is rational, it implies a form of periodicity in the multiplicative structure. The Riemann Hypothesis can be viewed as a statement about the maximum possible randomness between the additive structure of integers and their multiplicative structure. The fact that L_u remains in the rational class under specific conditions suggests a local-global principle: the global rationality of LN is entirely determined by local conditions at a finite set of primes (those p ≤ m+q). This mirrors the way local Euler factors determine the global behavior of the zeta function.

Sieve Bounds and Prime Density

The "gap" between a set LN and its nearest rational approximation can be quantified. If LN is not rational due to a missing prime p, then LN consists only of integers with specific p-adic valuations. This creates a sieve-like structure. The error term in the counting function for LN then becomes sensitive to the distribution of primes. The source paper's result that rationality is destroyed by the absence of even a single prime's powers is analogous to the sensitivity of the ζ(s) zero distribution to the values of the prime-counting function π(x).

Novel Research Pathways

Pathway 1: The Rationality Gap and Zero-Free Regions

We propose investigating the "Rationality Gap," defined as the minimum distance between the indicator function of LN and the space of rational sets in the Hardy space of Dirichlet series. If the Riemann Hypothesis holds, the density of primes is highly regular. We hypothesize that for sets L defined by prime properties, this gap can be bounded by a function of the supremum of the real parts of the non-trivial zeros of ζ(s). Researchers can use the "prime power saturation" theorem to construct a sequence of sets that approach a non-rational set and analyze the movement of the poles as the set loses its rationality.

Pathway 2: Automata Complexity and PNT Error Terms

Since rational sets are exactly those recognizable by finite automata, the complexity of a set LN can be measured by the number of states in its minimal recognizing automaton. The Prime Number Theorem (PNT) error term is bounded by x^1/2 log(x) under the Riemann Hypothesis. We propose that the state complexity of approximating the set of primes using rational sets scales at a rate determined by the width of the critical strip. This suggests a formal proof that the computational cost of representing multiplicative structure in an additive framework is dually related to the location of the zeta zeros.

Computational Implementation

The following Wolfram Language implementation demonstrates the rationality check for the multiplicative closure of a set L based on the criteria established in arXiv:hal-00019898v1. It evaluates prime power saturation and visualizes the density of the resulting set.

(* Section: Rationality and Density of Multiplicative Closures *)
(* Purpose: Check Theorem 6 of hal-00019898v1 and visualize prime distribution *)

Module[{L, m, q, primes, saturation, isRational, nMax, setLN, densityData},
  (* Define sample set L and parameters m, q *)
  L = {2, 3, 5, 9, 25};
  m = 2;
  q = 5;
  
  (* Get all primes up to m + q as per Theorem 6 *)
  primes = Table[Prime[k], {k, 1, PrimePi[m + q]}];
  
  (* Check if for each prime p, some power p^b is in L *)
  saturation = Table[
    AnyTrue[L, (IntegerQ[Log[primes[[i]], #]] && # > 0) &],
    {i, 1, Length[primes]}
  ];
  
  isRational = AllTrue[saturation, TrueQ];
  Print["Primes to check (m+q): ", primes];
  Print["Saturation Status: ", saturation];
  Print["Is L*N Rational? ", isRational];

  (* Visualize the density of L*N compared to Zeta behavior *)
  nMax = 500;
  setLN = Union[Flatten[Table[L * i, {i, 1, nMax}]]];
  setLN = Select[setLN, # <= nMax &];
  
  (* Calculate cumulative density *)
  densityData = Table[{x, Length[Select[setLN, # <= x &]] / x}, {x, 1, nMax}];
  
  Print[ListLinePlot[densityData, 
    PlotRange -> {0, 1}, 
    PlotLabel -> "Cumulative Density of L*N",
    AxesLabel -> {"x", "Density"}]];
    
  (* Compare with first few Zeta zeros *)
  Print["First 5 Non-trivial Zeta Zeros: ", Table[ZetaZero[n], {n, 1, 5}]]
]

Conclusions

The exploration of rational sets of integers through the lens of arXiv:hal-00019898v1 reveals that the algebraic properties of multiplicative closures are deeply intertwined with the distribution of prime numbers. The central finding—that the rationality of LN depends on the exhaustive presence of prime powers up to a specific bound—provides a discrete analog to the continuous requirements of the Riemann Hypothesis.

Our analysis suggests that the transition from rational to non-rational sets serves as a combinatorial marker for the complexity of the prime sequence. When a set fails the saturation test, its associated Dirichlet series escapes the realm of Hurwitz zeta functions and enters the critical strip where non-trivial zeros dictate its behavior. The most promising avenue for further research lies in quantifying this transition using automata theory and connecting state complexity to the width of the zero-free region of the Riemann zeta function.

References

• arXiv:hal-00019898v1: Rational sets of integers, 2006.
• Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press.
• Eilenberg, S. (1974). Automata, Languages, and Machines. Academic Press.
• Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.