Arithmetic Undecidability and the Analytic Bounds of Prime Density

Introduction

The boundary between decidable and undecidable mathematical structures is often defined by the presence of multiplication. While Presburger arithmetic (the theory of natural numbers with only addition) is decidable, the addition of a multiplication operation leads to the profound undecidability results of Gödel and Tarski. The research paper hal-00096769v1 explores a subtle middle ground: the existential theory of natural numbers equipped with addition and functions derived from the sequence of prime numbers. Specifically, it examines the structure (N, +, n ↦ p_n, n ↦ r_n), where p_n is the n-th prime and r_n is a remainder function.

The contribution of this analysis is to demonstrate that the logical complexity of this structure—specifically its undecidability—is not merely a combinatorial curiosity but is deeply rooted in the analytic regularity of prime numbers. The stability of the prime-quotient function, which allows for the definition of multiplication, is fundamentally constrained by the distribution of the zeros of the Riemann zeta function. By bridging the gap between mathematical logic and analytic number theory, we show how the Riemann Hypothesis (RH) provides the necessary "stiffness" in prime distribution to support these logical constructions.

Mathematical Background

The core mathematical object in hal-00096769v1 is the function f(n) = floor(p_n / n). According to the Prime Number Theorem, p_n is asymptotically equivalent to n log n, suggesting that f(n) behaves roughly like log n. However, the discrete nature of primes introduces local fluctuations. To manage these, the authors define a pseudo-inverse function f^-1(n), representing the smallest integer m such that f(m+1) > n.

A central property identified in the source paper is that f(n) is k-almost increasing. A function is k-almost increasing if, for all m > n, the value f(m) does not drop significantly below f(n). Specifically, the paper proves that for the prime-quotient function, f(m) ≥ f(n) - 1. This property, combined with lower bounds on the growth of the pseudo-inverse, allows for the existential definition of squaring and, subsequently, multiplication.

The connection to the Riemann zeta function, ζ(s), arises through the error terms of the prime counting function π(x). The Riemann Hypothesis asserts that all non-trivial zeros ρ of ζ(s) have a real part equal to 1/2. This hypothesis is equivalent to the tightest possible bound on the fluctuations of p_n, ensuring that the "noise" in the sequence does not overwhelm the logarithmic growth that the logical proofs rely upon.

Main Technical Analysis

Spectral Stability and the k-Almost Increasing Property

The undecidability proof in hal-00096769v1 depends on the regularity of the sequence floor(p_n / n). If the primes were distributed with extreme irregularity, the function f(n) would fail to be k-almost increasing, making it impossible to define the pseudo-inverse within the first-order language of addition. The source paper establishes that for n ≥ 11, the difference f^-1(n+1) - f^-1(n) is strictly greater than n.

Analytically, this regularity is a manifestation of the zero-free region of the Riemann zeta function. If ζ(s) had zeros with real parts significantly larger than 1/2, the resulting oscillations in the prime-counting function would manifest as large "dips" in the ratio p_n / n. Under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is O(n^1/2 log^2 n). This ensures that the derivative of the underlying trend log n + log log n - 1 dominates the oscillatory terms for sufficiently large n, preserving the almost-monotonic behavior required by the logic.

Defining Multiplication through Quadratic Growth

The gateway to undecidability is the definition of a function with quadratic growth. The source paper utilizes a telescoping sum of pseudo-inverse increments to establish the following inequality:
f^-1(n-1) - f^-1(n_0) > [(n-2)(n-1) - n_0(n_0+1)] / (2d)
where d is a fixed constant. This quadratic bound allows the structure to define the squaring function n ↦ n^2. Once squaring is available, multiplication is defined via the identity: xy = [(x+y)^2 - x^2 - y^2] / 2.

The paper also includes a computational verification for small values, noting that for k ≤ 7022, the maximum of p_k / k is attained at k = 7012 and remains below 10.103. This explicit check ensures that the logical construction is not invalidated by anomalous prime clusters in the early part of the sequence. Such bounds are consistent with the explicit formulas for primes derived from the first few thousand zeta zeros.

Novel Research Pathways

RH-Quantified Parameter Optimization

One promising direction is to determine the minimal parameters (k, d, n_0) for the class C(k, d, n_0) of functions that support undecidability. By assuming the Riemann Hypothesis, one could derive sharper explicit bounds for p_n, potentially reducing the starting index n_0 and the constant k. This would clarify the "logical complexity threshold"—the point at which the distribution of primes becomes regular enough to encode universal computation.

Spectral Diagnostics of Logical Plateaus

The plateaus where floor(p_n / n) remains constant are the inverse images of the pseudo-inverse function. We propose a research program to analyze the distribution of these plateau lengths as a discrete spectral signature. Under the Riemann Hypothesis, these lengths should follow a predictable distribution linked to the imaginary parts of the zeta zeros. Any significant deviation would indicate a violation of the critical line hypothesis and would simultaneously disrupt the existential definability of multiplication in the source paper's structure.

Computational Implementation

(* Section: Prime-Quotient Plateaus and Pseudo-Inverse Gaps *)
(* Purpose: To visualize the plateau structure of the prime density function 
   and verify the growth conditions required for logical undecidability. *)

Module[{nMax, f, fInv, gaps, zeros, tMax},
  nMax = 5000;
  tMax = 15;
  
  (* Define f(n) = floor(p_n / n) *)
  f[n_] := Floor[Prime[n]/n];
  
  (* Define pseudo-inverse fInv(t) as smallest m such that f(m+1) > t *)
  fInv[t_] := Module[{m = 1},
    While[f[m + 1] <= t && m < nMax, m++];
    m
  ];

  (* Calculate gaps in the pseudo-inverse *)
  gaps = Table[fInv[t + 1] - fInv[t], {t, 1, tMax}];

  (* Plot 1: The prime-quotient function showing plateau structure *)
  Print[ListLinePlot[Table[f[n], {n, 1, nMax}], 
    PlotLabel -> "Plateau Structure of Floor[p_n / n]", 
    AxesLabel -> {"n", "f(n)"}, PlotStyle -> Blue]];

  (* Plot 2: Pseudo-inverse gaps versus the linear bound n *)
  Print[ListPlot[gaps, 
    PlotLabel -> "Pseudo-Inverse Gaps f^-1(t+1) - f^-1(t)", 
    Filling -> Axis, AxesLabel -> {"t", "Gap Size"}]];

  (* Zeta zero connection: Display first 5 zeros *)
  zeros = Table[N[ZetaZero[k]], {k, 1, 5}];
  Print["First 5 Zeta Zeros (Critical Line): ", zeros];
  
  (* Final check: Value of f(n) vs Log[n] *)
  Print["Comparison at n=5000: f(5000) = ", f[5000], ", Log[5000] = ", N[Log[5000]]];
]

Conclusions

The analysis of hal-00096769v1 demonstrates that the sequence of prime numbers contains sufficient internal structure to define multiplication and thus inherit the undecidability of full arithmetic. This logical power is a direct consequence of the stability of prime density, which is in turn governed by the Riemann zeta function. The most promising avenue for future research lies in quantifying how the quantifier complexity of these logical formulas varies with the assumed zero-free region of the zeta function. By establishing that the Riemann Hypothesis is a necessary condition for certain types of arithmetic regularity, we may find new logical foundations for the most famous conjecture in number theory.

References

• hal-00096769v1: T. Lavendhomme and A. Maes, "Note on the undecidability of (N, +, P_m,r)".
• Rosser, J. B., and Schoenfeld, L. (1962). "Approximate formulas for some functions of prime numbers." Illinois Journal of Mathematics.
• Matiyasevich, Y. (1993). "Hilbert's Tenth Problem." MIT Press.