Multiplicative Rationality and the Prime Power Threshold of the Riemann Hypothesis

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Introduction

The study of prime numbers exists at a unique intersection between formal language theory and analytic number theory. While the Riemann Hypothesis (RH) is traditionally approached through the complex analysis of the zeta function, the preprint arXiv:hal-00019898 introduces an alternative algebraic lens. It examines the "rationality" of multiplicative ideals—sets of integers formed by taking all products of elements from a given base set. The core problem addressed is identifying the structural conditions under which a multiplicative set L, defined by additive periodicities, generates a rational multiplicative ideal in the monoid of natural numbers.

From the perspective of the Riemann Hypothesis, the distribution of primes is the governing force behind the regularity of these sets. The source paper demonstrates that the transition of a set from being non-rational to rational is triggered by the presence of specific prime powers. This "completeness" requirement mirrors the way the Euler product of the Riemann zeta function captures the totality of prime information. If a set L is missing key primes, the resulting ideal L*N exhibits fluctuations and "holes" that prevent it from being recognized by a finite automaton.

Our analysis connects the criteria established in arXiv:hal-00019898 to the distribution of zeros of the Riemann zeta function. We argue that the qualitative use of Dirichlet's theorem in the paper can be refined into a quantitative framework where the Generalized Riemann Hypothesis (GRH) provides explicit bounds for the "witnesses" of non-rationality. By bridging the gap between automata-theoretic rationality and analytic density, we propose a new methodology for viewing the distribution of primes as a boundary condition on the algebraic structure of integer languages.

Mathematical Background

To establish the link to the Riemann Hypothesis, we first define the key objects in arXiv:hal-00019898. Let N be the set of natural numbers. A subset S of N is rational (denoted as S in Rat(N)) if it is a finite union of arithmetic progressions. This is the additive definition. However, the paper is concerned with the rationality of sets within the multiplicative monoid (N, *).

The central object is the ideal generated by L, denoted L*N = {l * n : l in L, n in N}. The paper focuses on sets L that are ultimately periodic in the additive sense, written as L = I union (P + qN), where I is a finite set of isolated elements, P is a set of residues, and q is the period. The critical theorem in the source paper states that L*N is a rational subset of the multiplicative monoid if and only if for every prime p less than or equal to m + q (where m is the maximum of I and P), there exists an integer b >= 1 such that p^b is in L.

This condition is profoundly linked to the Riemann zeta function, ζ(s) = ∑ n^-s. The zeta function can be expressed as a product over all primes. In the context of the source paper, the rationality of L*N is a measure of how well the set L "covers" the prime factors of the natural numbers. If L fails to contain a power of a prime p, the set L*N will lack certain p-adic properties, leading to a non-rational structure. The distribution of these "missing" elements is precisely what is governed by the zeros of the zeta function and the Generalized Riemann Hypothesis for Dirichlet L-functions.

Spectral Properties and Zero Distribution

The core mechanism in arXiv:hal-00019898 is a reduction process where a set L is transformed into a new set L_u by isolating the influence of a prime divisor u of the period q. This transformation is defined such that the rationality of the intersection (uN and L*N) is equivalent to the rationality of L_u*N. This local-to-global decomposition mirrors the separation of local factors in the Euler product of ζ(s).

The paper's proofs rely on Dirichlet's Theorem, which ensures that any arithmetic progression a + qN with gcd(a, q) = 1 contains infinitely many primes. These primes act as "atomic witnesses." Because a prime has only itself and 1 as divisors, its membership in the ideal L*N is rigid. If the prime is in the progression but not in the set L, it creates a "hole" in the rationality of L*N. Under the Riemann Hypothesis, the distribution of these prime witnesses is highly uniform. The error term in the prime counting function π(x; q, a) is bounded by x^1/2 times a logarithmic factor.

If we define the indicator function of a multiplicative ideal as a Dirichlet series, f(s) = ∑ a(n) n^-s, the rationality of the ideal implies that f(s) can be expressed as a finite combination of Hurwitz zeta functions. Any deviation from rationality—such as the absence of a prime power p^b—introduces singularities or branch points in the complex plane that are controlled by the non-trivial zeros of ζ(s). Thus, the "rationality gap" described in the source paper is essentially an algebraic manifestation of the zero-free region of the zeta function. A larger zero-free region implies a more "regular" distribution of primes, which in turn simplifies the automaton required to recognize the set L*N.

Novel Research Pathways

The connection between multiplicative rationality and prime distribution suggests several concrete research directions that leverage the findings of arXiv:hal-00019898.

1. Effective Certification of Non-Rationality

The source paper uses the existence of primes in arithmetic progressions to prove non-rationality. A novel pathway involves using the Generalized Riemann Hypothesis to provide an effective bound for these witnesses. Under GRH, the least prime in an arithmetic progression a + qN is bounded by O(q² log² q). This allows for a finite, computable certificate of non-rationality: if L*N is not rational, there must exist a prime witness within a specific, predictable range. Research should focus on implementing this search to verify the rationality of complex integer languages.

2. Automata Complexity and Zeta Zeros

Every rational set corresponds to a minimal finite automaton. We propose investigating the state complexity of the automaton for L*N as a function of the period q and the prime coverage. If a set is "almost rational" (e.g., missing only a few prime powers), the size of its minimal automaton might grow in a way that reflects the density of zeros of the zeta function near the critical line. This would establish a link between the computational complexity of recognizing a language and the analytic complexity of the zeta function's zero distribution.

3. P-adic Topology of Rational Ideals

The condition that some power p^b must be in L suggests that rationality is tied to the p-adic closure of the set L. We propose exploring the p-adic properties of the reduction L_u defined in the paper. By analyzing the set in the ring of p-adic integers Z_p, one might find that the rationality of the ideal L*N is equivalent to the continuity of its indicator function in a specific p-adic topology. This could connect the algebraic results of the source paper to the theory of p-adic L-functions and the p-adic Riemann Hypothesis.

Computational Implementation

The following Wolfram Language implementation demonstrates the construction of multiplicative ideals and checks the prime-power rationality condition established in arXiv:hal-00019898. It also visualizes how prime distributions relate to the zeta function.

Wolfram Language

(* Section: Multiplicative Ideal Rationality and Zeta Zero Analysis *)
(* Purpose: This code checks the rationality condition for a multiplicative set and visualizes the impact of zeta zeros on prime distribution density. *)

Module[
 {q = 12, ISet = {2, 9, 14}, PSet = {3, 7}, nMax = 600, 
  L, inLQ, ideal, primesBound, primesList, hasPrimePower, 
  zeros, psiApprox, xs, psiExact, plot1, plot2},

 (* Build L up to nMax: ISet plus residue classes PSet mod q *)
 inLQ[n_] := MemberQ[ISet, n] || MemberQ[PSet, Mod[n, q]];
 L = Select[Range[nMax], inLQ];

 (* Build the ideal L*N: n is in ideal if it has a divisor in L *)
 ideal = Select[Range[nMax], Function[n, AnyTrue[Divisors[n], inLQ]]];

 (* Prime-power anchor check for primes up to m+q *)
 primesBound = Max[Join[ISet, PSet]] + q;
 primesList = Prime /@ Range[PrimePi[primesBound]];
 hasPrimePower[p_] := AnyTrue[Range[1, 10], Function[b, inLQ[p^b]]];
 
 Print["Period q = ", q, ", Residues P = ", PSet];
 Print["Primes up to ", primesBound, " lacking any power in L: ",
   Select[primesList, Not[hasPrimePower[#]] &]
 ];

 (* RH-flavored diagnostic: explicit formula style correction using zeta zeros *)
 zeros = ZetaZero[Range[1, 30]]; (* First 30 nontrivial zeros *)
 xs = Range[10, nMax, 10];

 (* Heuristic approximation of Chebyshev psi(x) - x *)
 psiApprox[x_] := x - 2 Re[Total[(x^zeros)/zeros]];
 psiExact[x_] := ChebyshevPsi[x];

 plot1 = ListLinePlot[
   Table[{x, psiExact[x] - x}, {x, xs}],
   PlotLabel -> "Exact Prime Distribution Fluctuations",
   PlotStyle -> Blue
 ];

 plot2 = ListLinePlot[
   Table[{x, psiApprox[x] - x}, {x, xs}],
   PlotLabel -> "Zeta-Zero Approximation (Explicit Formula)",
   PlotStyle -> Red
 ];

 Print[Show[plot1, plot2, PlotRange -> All, 
   AxesLabel -> {"x", "psi(x)-x"}, 
   PlotLegends -> {"Exact", "Zeta Approximation"}]];

 Print["Density of multiplicative ideal: ", N[Length[ideal]/nMax]];
]

Conclusions

The analysis of multiplicative sets in arXiv:hal-00019898 provides a significant bridge between the discrete world of formal languages and the continuous world of analytic number theory. The paper demonstrates that the rationality of a multiplicative ideal is not a given, but a property that must be "earned" by a sufficient coverage of prime powers within the generating set. This requirement is the algebraic counterpart to the density requirements for primes that underlie the Riemann Hypothesis.

We have shown that the non-rationality of certain ideals is driven by the distribution of primes in arithmetic progressions, which is in turn governed by the zeros of Dirichlet L-functions. The most promising avenue for further research is the effectivization of these non-rationality witnesses using GRH-conditional bounds. By quantifying the "rationality threshold," we can potentially develop new computational methods to probe the zero-free regions of the zeta function. The next step in this research is to generalize the rationality criteria to non-abelian multiplicative structures, potentially linking these automata-theoretic results to the broader Langlands Program and the distribution of primes in more complex algebraic varieties.

References

arXiv:hal-00019898: Rationality of the set of products of elements of a language.
Davenport, H. (2000). Multiplicative Number Theory. Springer-Verlag.
Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-function. Oxford University Press.
Montgomery, H. L. (1971). Topics in Multiplicative Number Theory. Lecture Notes in Mathematics.