The Log-Log Rhythm: Decoding Arithmetic Races in Prime Factorization and Zeta Zeros

Executive Summary

The research presented in arXiv:1806.01585v1 provides a rigorous analysis of the distribution of the additive function omega(n), representing the number of distinct prime factors of an integer n, within residue classes modulo q. By applying the Selberg-Delange method, the authors uncover persistent oscillatory behavior in the counting function N_a,q(x), which tracks how often omega(n) is congruent to a modulo q. These oscillations occur on a log-log scale (log log x), a phenomenon driven by the complex powers of the Riemann zeta function near its pole at s = 1.

The primary insight connecting this work to the Riemann Hypothesis (RH) lies in the precision of the error terms. While the main oscillatory term is deterministic and depends on the argument of a specific Euler product, the secondary fluctuations are sensitive to the zero-free regions of zeta(s). This article synthesizes the paper's findings to propose that these "arithmetic races" for omega(n) serve as a macroscopic mirror for the microscopic distribution of zeta zeros, offering a novel pathway to quantify zero-density estimates through the lens of additive prime statistics.

Introduction

Arithmetic statistics often exhibit "race" phenomena, where different counting functions compete for dominance. The most famous example is the Chebyshev bias, where primes congruent to 3 mod 4 appear more frequently than those congruent to 1 mod 4. In arXiv:1806.01585v1, the authors investigate a different kind of race: the distribution of the number of distinct prime factors omega(n) in residue classes modulo q. Unlike prime races, which are governed by the imaginary parts of zeros of L-functions, these races are governed by the singularity of the zeta function at s = 1.

The contribution of this analysis is twofold. First, it establishes that the difference between residue classes N_a,q(x) - N_b,q(x) changes sign infinitely often, driven by a slowly varying cosine wave. Second, it identifies the precise logarithmic densities of the sets of integers that favor one residue class over another. This structural behavior provides a bridge to the Riemann Hypothesis, as the stability and uniformity of these densities depend on the analytic continuation and zero-free regions of the zeta function.

Mathematical Background

The core mathematical object in this study is the summatory function A_z(x), defined as the sum over n up to x of z raised to the power of omega(n). When z is a q-th root of unity, this function allows for the extraction of residue classes modulo q via the discrete Fourier transform. According to arXiv:1806.01585v1, the asymptotic behavior of this sum is given by:

A_z(x) = g(z) x (log x)^(z-1) + O(x (log x)^(Re(z)-2))

The multiplicative constant g(z) is defined by the following Euler product:

• Definition: g(z) = (1/Gamma(z)) * Product over primes p of (1 + z/(p-1)) * (1 - 1/p)^z.
• Phase: The argument of g(phi), denoted as theta, where phi = exp(2*pi*i/q), determines the phase shift of the oscillations.
• Singularity: The term (log x)^(z-1) arises because the Dirichlet series of z^omega(n) behaves like zeta(s)^z near s = 1.

Under the Riemann Hypothesis, the remainder terms in these expansions can be significantly sharpened. RH implies that the non-trivial zeros rho lie on the critical line Re(s) = 1/2, which restricts the fluctuations of the error term to a magnitude of roughly x^(1/2). In the context of omega(n), this allows for a much tighter control over the "noise" that overlays the primary log log x oscillation.

Main Technical Analysis

Spectral Properties and log-log Oscillations

The counting function N_a,q(x) exhibits a periodic structure when viewed against the variable log log x. The source paper derives an explicit formula showing that the distribution of omega(n) mod q is not uniform but biased by a secondary term:

N_a,q(x) = (x/q) * (1 + 2|g(phi)| * cos(sin(2*pi/q) log log x + theta - 2*pi*a/q) * (log x)^(cos(2*pi/q)-1) + ...)

This formula demonstrates that the "race" between residue classes a and b is determined by the sign of the cosine term. Because the exponent cos(2*pi/q)-1 is negative, the amplitude of the oscillation decays as x increases, but the decay is extremely slow for large q. This slow decay makes the system highly sensitive to the distribution of primes and, by extension, the zeros of the zeta function.

Logarithmic Density and Phase Windows

To quantify this bias, arXiv:1806.01585v1 introduces sets U_a,q defined by phase windows. For a small epsilon, the set U_a,q (epsilon-) contains integers where the phase of the oscillation is such that the residue class a is significantly under-represented. The paper proves that the lower logarithmic density of these sets satisfies:

lim inf (1/log log X) * Sum_{n in U, n ≤ X} 1/(n log n) = 1/(2q) - sqrt(epsilon)/pi

The choice of the weight 1/(n log n) is crucial. It is the natural measure for the log log x scale, effectively turning the sum into an integral over the phase of the oscillation. This result shows that the bias is not a transient artifact but a persistent feature of the integers, with a density that is strictly less than the expected 1/q.

Novel Research Pathways

1. Explicit Formulas for Additive Function Remainder Terms

A promising research direction is to develop an "explicit formula" for A_z(x) that directly incorporates the non-trivial zeros of the zeta function. While the Selberg-Delange method focuses on the singularity at s = 1, a more refined approach would involve shifting the contour of the Perron integral to the critical line. Under RH, this would yield a sum over zeros rho of the form Sum x^rho / rho, modulated by the g(z) product. This would allow researchers to see how individual zeta zeros contribute to the "jitter" in the omega(n) distribution.

2. The Rubinstein-Sarnak Distribution for omega(n)

In the theory of prime number races, Rubinstein and Sarnak established the existence of a limiting distribution for the error term. A similar framework could be applied here. By normalizing the difference N_a,q(x) - N_b,q(x) by its expected amplitude (x/q)(log x)^(cos(2*pi/q)-1), one could investigate whether the residual fluctuations follow a specific probability distribution. The shape of this distribution (e.g., its symmetry or lack thereof) would be a direct consequence of the Linear Independence (LI) hypothesis of the imaginary parts of zeta zeros.

3. Transition Regimes for Large q

As q grows, the term sin(2*pi/q) approaches zero, meaning the period of the oscillation in log log x becomes increasingly long. Investigating the transition where q is a function of x (e.g., q ∼ sqrt(log log x)) would reveal a regime where the deterministic oscillation and the stochastic error from zeta zeros are of comparable magnitude. This "interference" zone could provide new bounds on the growth of the zeta function on the critical line.

Computational Implementation

(* Section: Omega(n) mod q Race Visualization *)
(* Purpose: Compare empirical bias to the theoretical cosine wave *)

Module[{q = 3, a = 1, limit = 5000, phi, gVal, theta, absG, data, theoryPlot, empiricalPlot},
  phi = Exp[2. * Pi * I / q];
  
  (* Approximate g(phi) using first 200 primes *)
  gVal = (1/Gamma[phi]) * Product[
    (1 + phi/(Prime[p] - 1)) * (1 - 1/Prime[p])^phi, 
    {p, 1, 200}
  ];
  theta = Arg[gVal];
  absG = Abs[gVal];
  
  (* Calculate empirical N_a,q(x) - x/q *)
  data = Table[
    {x, Count[Mod[PrimeOmega[Range[x]], q], a] - x/q},
    {x, 100, limit, 100}
  ];
  
  (* Theoretical oscillatory term *)
  theoryPlot = Plot[
    (x/q) * 2 * absG * Cos[Sin[2*Pi/q] * Log[Log[x]] + theta - (2*Pi*a)/q] * 
    (Log[x])^(Cos[2*Pi/q] - 1), 
    {x, 100, limit}, 
    PlotStyle -> Red, PlotLabels -> "Theory"
  ];
  
  empiricalPlot = ListLinePlot[data, PlotStyle -> Blue, PlotLabels -> "Empirical"];
  
  Show[empiricalPlot, theoryPlot, 
    AxesLabel -> {"x", "Bias"}, 
    PlotLabel -> "Log-Log Oscillation in Omega(n) mod 3"]
]

The code above demonstrates the slow-moving bias predicted by arXiv:1806.01585v1. By comparing the blue empirical line with the red theoretical curve, one can observe the "arithmetic race" in action. The divergence between the two at small x is exactly where the influence of zeta zeros and the secondary terms of the Selberg-Delange method are most prominent.

Conclusions

The study of omega(n) in residue classes modulo q reveals a structured, oscillatory universe within the integers. The findings in arXiv:1806.01585v1 prove that these distributions are not random but follow a deterministic rhythm set by the values of the zeta function near the line Re(s) = 1. The existence of these log-log races confirms that additive properties of primes are subject to the same transcendental biases found in multiplicative number theory.

The most promising avenue for future work is the integration of these macroscopic oscillations with the microscopic data of the zeta zeros. Establishing a formal connection between the variance of these arithmetic races and the Riemann Hypothesis would provide a powerful new tool in the quest to prove that all non-trivial zeros lie on the critical line.

References

• arXiv:1806.01585v1: Oscillatory behavior of the number of distinct prime factors in residue classes.
• Selberg, A. (1954). "Note on a paper by L. G. Sathe". Journal of the Indian Mathematical Society.
• Tenenbaum, G. (2015). "Introduction to Analytic and Probabilistic Number Theory". Cambridge University Press.