Mapping the Critical Line Through Logarithmic Mantissas and Factorization Tables

Introduction

The distribution of prime numbers remains one of the most enduring mysteries in mathematics, fundamentally tied to the zeros of the Riemann zeta function, ζ(s). Historically, progress in this field has relied on the construction of extensive numerical tables to verify conjectures and identify patterns. The source paper arXiv:hal-00654431 continues this tradition by providing a rigorous compilation of prime factorizations and discrete logarithms (indices) for integers within specific ranges, specifically organized by residue classes modulo 10.

While such tables are often viewed as tools for modular arithmetic or cryptography, they provide the raw data necessary to investigate the fluctuations of the error term in the prime counting function, π(x). This analysis interprets the structured data in arXiv:hal-00654431 as a computational artifact perfectly suited for evaluating Chebyshev functions and Dirichlet L-functions. By bridging the gap between the discrete factorizations and the continuous analytic properties of the zeta function, we can develop new methodologies for testing the Riemann Hypothesis (RH) through the structural regularities of prime-admissible residue classes.

Mathematical Background

The connection between primes and the zeros of the Riemann zeta function is formalized by the Explicit Formula. This formula relates the Chebyshev function ψ(x), defined as the sum of the von Mangoldt function Λ(n) for n less than or equal to x, to the non-trivial zeros ρ of ζ(s). The Riemann Hypothesis asserts that all these zeros have a real part of exactly 1/2, which implies that the error in the prime distribution is on the order of x^1/2 log² x.

The tables in arXiv:hal-00654431 focus on the residue classes 1, 3, 7, and 9 modulo 10. These are the only possible terminal digits for primes greater than 5. Primes in these arithmetic progressions are governed by Dirichlet L-functions L(s, χ). The Generalized Riemann Hypothesis (GRH) extends RH by predicting that the zeros of these L-functions also lie on the critical line. The dataset in the source paper allows for a precise evaluation of the weighted prime-counting function θ(x; 10, a), which sums log p for primes p congruent to a mod 10. Because the building blocks of analytic number theory are logarithmic weightings, the high-precision base-10 mantissas provided in the paper are directly relevant to verifying square-root cancellation in prime races.

Main Technical Analysis

Hybrid Logarithmic-Factorial Structures

A central feature of arXiv:hal-00654431 is its hybrid entry system. For a given integer block, the tables record either a prime factorization (e.g., 18001 = 47 * 383) or a 10-digit integer representing a common logarithm mantissa (e.g., for prime 18041, the entry is 2562606065). This dual structure serves as a compressed logarithm table. If n is composite, its logarithm is the sum of the logarithms of its factors; if n is prime, its logarithm is stored directly.

This organization allows researchers to reconstruct the natural logarithm log p with high fidelity using the identity log(p) = ln(10) * log10(p). Since the integer part of log10(p) is determined by the number of digits, the tabulated mantissas provide the necessary precision to calculate the Chebyshev function θ(x) = Σ log p. Any deviation from the expected density of primes in these columns would signal a clustering that contradicts the spectral repulsion predicted by the Montgomery-Odlyzko law.

Spectral Properties and Residue Distribution

The Montgomery-Odlyzko law suggests that the spacings between zeta zeros behave like the eigenvalues of random Hermitian matrices. By mapping the indices and logarithms from arXiv:hal-00654431 into frequency domains, we can analyze the local spectral density of primes. The residue class organization (1, 3, 7, 9) is particularly significant for studying "prime number races." Under GRH, the difference between the number of primes in different residue classes should oscillate within a predictable envelope. The source paper's data provides a vetted baseline for testing these oscillations and identifying any subtle biases that could indicate zeros off the critical line.

Novel Research Pathways

1. Log-Weighted Prime Races mod 10

Formulation: Define W_a(x) = θ(x; 10, a) - x/4. Under GRH, one expects W_a(x) to scale with x^1/2 log² x. Using the precise log p data from the tables, researchers can compute the normalized process Z_a(x) = W_a(x) / (x^1/2 log x) and study its covariance structure. This pathway uses the multi-progression structure to enhance signal detection for zeta zero imaginary parts.

2. Euler-Product Truncation Diagnostics

Formulation: Approximate ζ(s) near the critical line using the truncated product over primes p less than P. The stable logs provided in arXiv:hal-00654431 allow for more accurate computation of p^-s = exp(-s log p). By correlating truncation errors with the proximity to known zeta zeros, we can develop quantitative "phase diagrams" of how the distribution of zeros constrains prime-based approximations.

3. Mantissa Randomness and Digital Distribution

Formulation: Analyze the statistical distribution of the fractional parts of log10(p) within each residue class. Discrepancies from uniform distribution can be expressed as exponential sums over primes. This research would investigate whether the base-10 digital structure of primes, as recorded in the source paper, exhibits statistical signatures compatible with the GUE predictions of random matrix theory.

Computational Implementation

The following Wolfram Language implementation demonstrates how to extract prime distribution data from specific ranges and correlate the gaps between primes with the spacings of zeta zeros, reflecting the spectral analysis suggested by the source paper.

(* Section: Prime Gap Analysis and Zeta Zero Correlations *)
(* Purpose: Compare arithmetic progression gaps to RH predictions *)

Module[{
  start = 18000, end = 20000, 
  primes, gaps, zetaZeros, 
  plot1, plot2
},
  (* Select primes in the range consistent with arXiv:hal-00654431 *)
  primes = Select[Range[start, end], PrimeQ];
  
  (* Calculate normalized gaps: g_n = (p_{n+1} - p_n) / log(p_n) *)
  gaps = Table[
    (primes[[i + 1]] - primes[[i]]) / Log[primes[[i]]], 
    {i, 1, Length[primes] - 1}
  ];
  
  (* Extract imaginary parts of the first 50 zeta zeros *)
  zetaZeros = Table[Im[ZetaZero[n]], {n, 1, 50}];
  
  (* Visualization 1: Histogram of normalized gaps *)
  plot1 = Histogram[gaps, {0.2}, "Probability", 
    PlotLabel -> "Normalized Prime Gaps (18000-20000)", 
    AxesLabel -> {"Gap Size", "Frequency"}];
    
  (* Visualization 2: Spacings of Non-Trivial Zeros *)
  plot2 = ListLinePlot[Differences[zetaZeros], 
    PlotLabel -> "Zeta Zero Spacings", 
    PlotStyle -> Red];
    
  (* Output statistics *)
  Print["Number of primes analyzed: ", Length[primes]];
  Print["Mean normalized gap: ", Mean[gaps]];
  Print[GraphicsRow[{plot1, plot2}]]
]

Conclusions

The analysis of arXiv:hal-00654431 demonstrates that high-precision tables of factorizations and logarithms are fundamentally linked to the analytic structure of the Riemann zeta function. The organization of data into residue classes mod 10 aligns with the Dirichlet decomposition of primes, providing a unique dataset for testing GRH-calibrated envelopes. The most promising future direction involves the spectral analysis of these discrete logarithms to prove they follow the spacing distributions predicted by random matrix theory. Expanding these tables to higher ranges of x will remain a vital tool in verifying the location of zeros on the critical line and resolving the Riemann Hypothesis.

References

• Original Source: arXiv:hal-00654431
• Montgomery, H. L. (1973). "The pair correlation of zeros of the zeta function." Proc. Sympos. Pure Math.
• Odlyzko, A. M. (1987). "On the distribution of spacings between zeros of the zeta function." Mathematics of Computation.
• Rubinstein, M., and Sarnak, P. (1994). "Chebyshev's bias." Experimental Mathematics.