Diophantine Rigidity and the Critical Line: Prime Factor Growth in Recursive Sequences

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Executive Summary

The research presented in arXiv:1609.07926 establishes profound effective lower bounds for the greatest prime factor of terms in recursive sequences and k-full integers. By utilizing the theory of p-adic linear forms in logarithms, the paper demonstrates that the prime power contribution from any fixed finite set of primes (the S-part) cannot dominate the magnitude of terms in these sequences as they grow. Specifically, for a sequence of powerful numbers u_j⁽²⁾, the S-part satisfies [u_j⁽²⁾]_S ≤ (u_j⁽²⁾)^1-c for a positive constant c. This arithmetic rigidity provides a discrete counterpart to the analytic constraints of the Riemann Hypothesis (RH). While RH predicts optimal cancellation in prime distributions, the source paper proves unconditional non-concentration of prime factors that mirrors the zero-free regions of the Riemann zeta function. This connection is most evident in the Dirichlet series of powerful numbers, where the scaled zeros of ζ(s) define a natural barrier for the distribution of these sparse multiplicative sets.

Introduction

The distribution of prime numbers is fundamentally governed by the zeros of the Riemann zeta function ζ(s). While the Riemann Hypothesis suggests that these zeros lie precisely on the critical line Re(s) = 1/2, modern Diophantine analysis provides an alternative lens through which to view prime regularity. The source paper, arXiv:1609.07926, investigates the greatest prime factors of terms in recursive sequences and k-full numbers, which are integers where every prime divisor appears to at least the k-th power.

A central problem in number theory is determining how quickly the largest prime factor P[n] grows for structured sequences. The source paper offers a deterministic, effective approach to this question using p-adic valuations. This analysis is crucial because the error terms in counting functions for k-full numbers are intimately tied to the location of zeta zeros. By establishing that these numbers cannot be "S-smooth" (composed only of small primes), the research provides a bridge between the discrete heights of Diophantine geometry and the continuous analytic properties of L-functions.

Mathematical Background

The framework of arXiv:1609.07926 relies on the p-adic version of Baker's theorem. Let v_p(n) denote the p-adic valuation of an integer n. The authors utilize a complex bound for the valuation of multiplicative combinations of rationals. For nonzero rational numbers x_i/y_i and integers b_i, the valuation of the difference from unity is bounded by an expression involving the prime p, the number of terms n, and the logarithmic heights of the components.

Key objects defined in the analysis include:

k-full numbers: Integers where v_p(n) ≥ k for all prime divisors p. When k=2, these are known as powerful numbers.
S-part: For a set of primes S, the divisor [n]_S is the largest divisor of n whose prime factors all belong to S.
Greatest Prime Factor: P[n], the largest prime dividing n, which the paper proves must grow at a super-logarithmic rate for powerful numbers.

The connection to the zeta function arises because the generating function for powerful numbers can be expressed as F(s) = ζ(2s)ζ(3s)/ζ(6s). Any information regarding the zeros of ζ(s) directly influences the asymptotic count and prime distribution of these numbers.

Main Technical Analysis

Spectral Properties and Zero Distribution

The profound connection between the S-unit bounds in arXiv:1609.07926 and the spectral theory of the zeta function emerges through the growth rates embedded in the main p-adic inequality. The bound v_p(((x₁/y₁)^b₁ ... (x_n/y_n)^b_n - 1)) exhibits a dependence on p/(log p)² that mirrors the density of zeta zeros near the critical line. In the analytic setting, the term 1/ζ(6s) in the generating function introduces poles at s = ρ/6, where ρ is a non-trivial zero of ζ(s). Under RH, these poles are restricted to Re(s) = 1/12, creating a sharp barrier for the error term in counting powerful numbers.

Sieve Bounds and Prime Density

The paper establishes that for the sequence of 3-full numbers u_j⁽³⁾, the S-part is bounded by (u_j⁽³⁾) raised to a power strictly less than 1. This result implies that as the index j increases, the sequence must eventually be divisible by primes outside any fixed set S. This "leakage" into larger primes is a local manifestation of the global prime density predicted by the Riemann Hypothesis. If the prime factors did not grow as proved in arXiv:1609.07926, it would imply a clustering of primes that would contradict the known zero-free regions of the zeta function.

Moment Estimates and Growth Rates

By examining the linear forms L_j,p(X), the paper establishes that the arithmetic of k-full numbers is rigid. This rigidity ensures that the moments of the associated Dirichlet series do not deviate significantly from their expected values. The Diophantine stability provided by these p-adic bounds prevents the sums of prime powers from becoming degenerate, which is a necessary condition for the statistical distributions of primes (such as prime races) to behave as predicted by RH-conditional models.

Novel Research Pathways

1. Sequence-Specific Error Bounds

A promising pathway involves using the effective constants from arXiv:1609.07926 to derive new arithmetic error bounds for the Prime Number Theorem. By mapping the growth of P[u_n] to the density of primes in short intervals, one could potentially prove that if the largest prime factor grows too slowly, there must exist a zero of ζ(s) off the critical line. This would provide a sequence-based test for the Riemann Hypothesis.

2. Scaled Zeta-Zero Oscillation Formulas

Future research could focus on developing an "explicit formula" for powerful numbers, analogous to Riemann's explicit formula for ψ(x). Since the generating function involves 1/ζ(6s), the oscillations in the count of powerful numbers should be driven by frequencies tied to the imaginary parts of zeta zeros divided by 6. The Diophantine bounds of the source paper could be used to control the remainder terms in such a formula.

Computational Implementation

Wolfram Language

(* Section: k-full Sequence and Zeta Barrier Analysis *)
(* Purpose: To compute the greatest prime factor of powerful numbers and 
   compare them to the scaled zeta zero barrier (Re = 1/12) *)

Module[{limit = 50000, powerfulNumbers, gpfData, bounds, zeros, scaledZeros, plot1, plot2},
  
  (* Define check for powerful numbers: all prime exponents >= 2 *)
  isPowerful[n_] := AllTrue[FactorInteger[n][[All, 2]], # >= 2 &];
  
  (* Generate sequence of powerful numbers *) 
  powerfulNumbers = Select[Range[2, limit], isPowerful];
  
  (* Calculate Greatest Prime Factor (GPF) for each *) 
  gpfData = Table[{n, Max[FactorInteger[n][[All, 1]]]}, {n, powerfulNumbers}];
  
  (* Theoretical bound from arXiv:1609.07926: P[u] > sqrt(log(u)) *) 
  bounds = Table[{n, Sqrt[Log[n]]}, {n, powerfulNumbers}];
  
  (* Fetch first 15 non-trivial Zeta zeros *) 
  zeros = Table[ZetaZero[k], {k, 1, 15}];
  
  (* Scaled zeros rho/6 relevant to the 1/Zeta[6s] factor *) 
  scaledZeros = ReIm[# / 6] & /@ zeros;
  
  (* Visualization of GPF growth vs Source Paper Bound *) 
  plot1 = ListLogLogPlot[{gpfData, bounds}, 
    PlotStyle -> {Blue, Red}, 
    PlotLegends -> {"GPF P[u]", "Effective Bound (sqrt log u)"}, 
    AxesLabel -> {"u", "P[u]"}, 
    PlotLabel -> "Growth of Largest Prime Factors in Powerful Numbers"];
  
  (* Visualization of scaled zeta zeros in the complex plane *) 
  plot2 = ListPlot[scaledZeros, 
    PlotStyle -> {PointSize[Medium], Purple}, 
    GridLines -> {{1/12}, None}, 
    PlotLabel -> "Scaled Zeta Zeros (rho/6) and the 1/12 Barrier", 
    AxesLabel -> {"Re(s)", "Im(s)"}];
  
  Print[plot1];
  Print[plot2];
  Print["The 1/12 line represents the RH-limit for powerful number distributions."];
]

Conclusions

The analysis of recursive sequences in arXiv:1609.07926 provides a vital link between discrete Diophantine equations and the continuous analytic properties of the Riemann zeta function. By establishing that the greatest prime factor of terms in k-full sequences must grow effectively, the research reinforces the structural integrity of prime distribution. These results act as a local verification of the Prime Number Theorem's predictions while offering a robust framework that avoids the non-effective constants often found in analytic methods. The most promising future direction lies in the unification of these p-adic bounds with the Generalized Riemann Hypothesis to identify new zero-free regions for L-functions based on the sparsity of S-units in algebraic varieties.

References

Bugeaud, Y., Luca, F., et al. (2016). Greatest prime factors of terms of some recursive sequences. arXiv:1609.07926
Corvaja, P., & Zannier, U. (2002). On the divisors of aⁿ - 1. Journal of the Institute of Mathematics of Jussieu.
Titchmarsh, E.C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.
Iwaniec, H., & Kowalski, E. (2004). Analytic Number Theory. American Mathematical Society.