Prime Radical Dynamics and the Riemann Critical Strip: New Theoretical Pathways

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Introduction

The distribution of prime numbers is a fundamental problem in mathematics, traditionally addressed through the analytic properties of the Riemann zeta function. However, recent developments in arithmetic geometry and additive number theory, particularly surrounding the ABC conjecture, have suggested deeper connections between the multiplicative structure of integers and the density of primes. The source paper arXiv:hal-01770397, entitled "Around the ABC Conjecture," provides a unique framework for exploring these connections through the lens of the radical function.

The radical of an integer, defined as the product of its distinct prime factors, serves as a bridge between the additive properties of sums and the multiplicative properties of prime decomposition. By investigating inequalities that involve the radical of an integer, the source paper establishes a series of bounds that intersect with the prime counting function pi(x) and the Chebyshev function psi(x). This article provides a comprehensive technical analysis of how these radical-based structures might inform our understanding of the Riemann Hypothesis and the behavior of zeros on the critical line.

The central motivation of this analysis is to determine whether the epsilon-dependent bounds found in arXiv:hal-01770397 can be used to refine the zero-free regions of the zeta function. We explore the possibility of a Radical-Zeta Correspondence, where the growth of the radical function acts as a regulator for the oscillations of the prime-counting error term.

Mathematical Background

To analyze the connections presented in arXiv:hal-01770397, we first define the core mathematical objects. The radical of a positive integer n, denoted by rad(n), is the product of the distinct primes dividing n. For any coprime triple (a, b, c) where a + b = c, the ABC conjecture posits that for any epsilon > 0, there exists a constant C(epsilon) such that c is bounded by C(epsilon) times rad(abc) raised to the power of 1 + epsilon.

The source paper introduces a parameter pi_p, which denotes a prime-counting or prime-index quantity, and a scaling variable K. These are used to construct inequalities such as:

log(pi_p / rad^{1/2(1+epsilon)}) < 1 + 1/K
|log(K)| ≤ 1 / (K N_f)

The presence of the 1/2 power in the radical term is significant. In the context of the Riemann Hypothesis, the square-root of x is the natural scale for the error term in the Prime Number Theorem. Specifically, the Riemann Hypothesis is equivalent to the statement that the difference between the Chebyshev function psi(x) and x is bounded by a constant times the square root of x times the square of the logarithm of x.

The paper also discusses the complex-valued radical function rad^s where s = alpha + i beta. When the real part alpha is restricted to the interval (0, 1), the analysis moves directly into the critical strip of the complex plane, where the non-trivial zeros of the zeta function are located.

Main Technical Analysis

Spectral Properties and Radical Thresholds

The core analysis in arXiv:hal-01770397 involves a sequence of transformations that bound the radical of an integer against its prime density. A representative inequality from the paper describes a threshold behavior:

rad^{2 epsilon rad^{1/2(1+epsilon)} - pi_p epsilon²} 4^{2 pi_p(1+epsilon)} > 1

This expression suggests that the distribution of prime factors is constrained by an exponential growth law. In analytic number theory, the stability of the zeta function's zeros depends on sums over primes not fluctuating beyond specific bounds. The term rad^{1/2(1+epsilon)} acts as a regulator. If we interpret this inequality through the explicit formula, it provides a constraint on how often prime powers can concentrate near specific intervals.

The paper identifies a "meeting place" of multiplicative information through Least Common Multiple (LCM) constructions. By expanding x_1 = LCM(a - 1, b - 1) times pi, the author factors out the radical, showing that the additive shift (p - 1) imports information from the radical into the prime distribution. This is a common theme in sieve theory, where the distribution of shifted primes is linked to the density of zeros of Dirichlet L-functions.

Sieve Bounds and Local Prime Density

Structure 13 of the source paper provides a complex bound involving the integral of a function P(t) and the prime count of the square root of c. This mirrors the logic of the Selberg Sieve, where one seeks to bound the number of integers in a set that are not divisible by any prime in a given range. The appearance of the square root of c is essential; it confirms that the radical-based approach naturally respects the square-root law of prime fluctuations.

The paper suggests that the "deficit" between an integer and its radical (n - rad(n)) can be used to bound the remainder term in these sieve estimates. If the radical function can be used to constrain the remainder, it would provide a direct path to proving that the fluctuations of psi(x) are controlled, thereby validating the Riemann Hypothesis's prediction for the zero distribution.

Novel Research Pathways

Pathway 1: The Radical Zeta Correspondence

We propose the formalization of a Radical Zeta Function, defined as the sum of 1 / rad(n)^s for all positive integers n. While the standard zeta function sums over n^-s, this modified function weights integers by their distinct prime factors. Since rad(n) is always less than or equal to n, this series diverges more slowly but contains concentrated information about prime support.

Formulation: Analyze the ratio of zeta(s) to the Radical Zeta Function in the critical strip.
Connection: Use the radical inequalities from arXiv:hal-01770397 to prove that the Radical Zeta Function has no zeros for Re(s) > 1/2.
Methodology: Apply the epsilon-scaling method to the coefficients of the Dirichlet series to ensure stability near the critical line.

Pathway 2: Epsilon-Controlled Zero Approximation

The source paper's use of the epsilon parameter suggests a method for approximating zero locations. By constructing a family of Diophantine equations parameterized by epsilon, one might characterize the critical line as the limiting case where these equations have solutions.

Formulation: Define a threshold function delta(epsilon) that relates the radical growth to the height of zeros T.
Expected Outcome: A theorem establishing that as epsilon approaches zero, the solutions to the radical constraints cluster around the imaginary parts of the non-trivial zeros.

Computational Implementation

To investigate the relationship between radical growth and zeta zero distribution, we implement a Wolfram Language script. This code calculates the cumulative radical sum and compares it to the predicted density of primes and the oscillations of zeta zeros.

(* Section: Radical and Zeta Zero Correlation *)
(* Purpose: Visualize the error term oscillation against radical deficit *)

Module[{maxN = 1000, rad, psiExact, nValues, psiErr, radDeficit, zeros, plot},
  (* Define the Radical function *)
  rad[n_] := Times @@ FactorInteger[n][[All, 1]];
  
  (* Define Exact Chebyshev Psi error *)
  psiExact[x_] := ChebyshevPsi[x];
  
  nValues = Range[2, maxN];
  
  (* Calculate the error term (psi(x) - x) *)
  psiErr = Table[psiExact[x] - x, {x, nValues}];
  
  (* Calculate the radical deficit normalized for visualization *)
  radDeficit = Table[(x - rad[x])/x, {x, nValues}];
  
  (* Obtain the first few Zeta Zeros for comparison *)
  zeros = Table[Im[ZetaZero[k]], {k, 1, 10}];
  
  (* Plot the results *)
  plot = ListLinePlot[{psiErr, radDeficit * 100},
    PlotLegends -> {"Psi Error (psi(x)-x)", "Scaled Radical Deficit"},
    AxesLabel -> {"n", "Magnitude"},
    PlotLabel -> "Radical Deficit vs. Prime Distribution Error",
    PlotStyle -> {Blue, Red}
  ];
  
  Print[plot];
  Print["First 10 Zeta Zero Heights: ", zeros];
]

Conclusions

The analysis of arXiv:hal-01770397 reveals a sophisticated interplay between the radical of an integer and the density of prime numbers. By deriving logarithmic inequalities that bound the radical function, the paper provides a framework that is remarkably consistent with the analytic requirements of the Riemann Hypothesis. Specifically, the emergence of the 1/2 power in the radical bounds mirrors the square-root error term necessary for the hypothesis to hold on the critical line.

The most promising avenue for further research lies in the exploration of the Radical Zeta Function within the critical strip. If the combinatorial constraints on the radical can be translated into analytic constraints on Dirichlet series, it may be possible to show that the zeros of the zeta function are forced onto the critical line by the additive-multiplicative rigidity of the integers. Future work should focus on applying the sieve-theoretic bounds from the paper to the explicit formula for the prime-counting function.

References

arXiv:hal-01770397 - Around the ABC Conjecture.
Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse.
Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function.
Iwaniec, H., and Kowalski, E. (2004). Analytic Number Theory.