Modular Prime Sieve: Investigating the Riemann Hypothesis through Last-Digit Composite Families

Introduction

The distribution of prime numbers is fundamentally linked to the behavior of composite numbers. In the research paper arXiv:hal-01407334v1, a systematic approach is presented to classify all odd composite numbers by their terminal digits in base 10. By partitioning odd integers into five residue classes modulo 10—those ending in 1, 3, 5, 7, or 9—the author identifies fifteen distinct product families that generate the entire set of composite odd integers.

While the Riemann Hypothesis is usually approached through complex analysis and the zeros of the zeta function, this residue-based classification offers a combinatorial bridge. By understanding the structured "holes" in the number line created by these fifteen composite families, we can analyze the resulting prime density. This article explores how the algebraic structures in the source paper relate to Dirichlet L-functions and the spectral properties of prime fluctuations on the critical line.

Mathematical Background

The core of the analysis in arXiv:hal-01407334v1 rests on the unit group of integers modulo 10. Every odd number n lies in one of five residue classes: S₁, S₃, S₅, S₇, or S₉. Because the product of any two odd numbers is also odd, the set of all composite odd numbers is the union of the products of these five sets.

The paper defines fifteen families, P₁ through P₁₅, representing all possible pairings. For example:

• P₁: (10n₁ + 1)(10n₁ + 1) = 10(10n₁² + 2n₁) + 1
• P₈: (10n₂ + 3)(10n₄ + 7) = 10(10n₂n₄ + 7n₂ + 3n₄ + 2) + 1
• P₁₅: (10n₅ + 9)(10n₅ + 9) = 10(10n₅² + 18n₅ + 8) + 1

These equations provide a deterministic way to identify composite numbers. The Riemann Hypothesis is equivalent to the statement that the error term in the prime counting function π(x) is bounded by x^1/2 log x. In this context, the error is determined by the regularity of the overlaps and gaps between these fifteen arithmetic families.

Main Technical Analysis

Sieve Bounds and Prime Density

The fifteen equations act as a "composite selector." If we consider a large interval [1, x], the number of primes π(x) is simply the number of odd integers minus the cardinality of the union of the sets P_i. This modular sieve partitions the "parity problem" into localized residue classes. For instance, the set S₅ (numbers ending in 5) is composed entirely of composites except for the number 5 itself. The remaining four classes (1, 3, 7, 9) contain all other primes.

The paper's empirical data tracks the evolution of the cumulative number of primes over bins of size 10⁵. These counts show fluctuations that are not purely random. According to the explicit formula of analytic number theory, these fluctuations are a superposition of waves whose frequencies are determined by the imaginary parts of the nontrivial zeros of the Riemann zeta function ζ(s).

Spectral Properties and Zero Distribution

Each composite family P_i contributes a specific density to the number line. When these densities are combined, they create an interference pattern. If the Riemann Hypothesis holds, the primes must be distributed with a specific square-root variance. In the source paper, the binned prime counts exhibit a "spectral regularity" that suggests the underlying composite generators are balanced in a way that prevents zero-clustering away from the critical line Re(s) = 1/2.

By analyzing the representation function r(n), which counts how many ways an integer n can be formed by the fifteen product families, we can relate the sieve's complexity to the Mobius function μ(n). The summatory function of μ(n) is the primary tool for proving the Riemann Hypothesis; here, it is interpreted as the alternating sum of composite multiplicities within the residue classes.

Novel Research Pathways

Pathway 1: Structured Hyperbola Sieves

One can treat the fifteen families as a system of shifted hyperbolas. By defining sieve weights derived from the bilinear forms in arXiv:hal-01407334v1, researchers can seek to bound the character sums for Dirichlet L-functions modulo 10. A proof that these weights provide square-root cancellation would be equivalent to a localized version of the Generalized Riemann Hypothesis.

Pathway 2: Dirichlet-Series Encoding

Each family P_i can be associated with a Dirichlet series F(s). Since these families are products of linear forms, their generating functions can be decomposed into products of Hurwitz zeta functions ζ(s, a/10). Analyzing the analytic continuation of these series allows us to study the "residue-class entropy," providing a new metric for the randomness of prime gaps.

Pathway 3: p-adic Generalizations

The base-10 structure is specific to our decimal system, but the logic extends to any prime base p. By comparing the fifteen families in base 10 to the analogous structures in base 2, 3, or 7, we can look for universal constants in the fluctuation patterns. This p-adic approach might reveal why the zeta zeros are forced onto the critical line by the requirements of modular symmetry.

Computational Implementation

(* Section: Spectral Analysis of Modular Prime Fluctuations *)
(* Purpose: This code compares binned prime counts in residue classes *)
(* modulo 10 against an oscillatory model derived from zeta zeros. *)

Module[{limit = 10^6, binSize = 5*10^4, xVals, residues, piData, liBase, deviations, imZeros, model, plot1},
  xVals = Range[binSize, limit, binSize];
  residues = {1, 3, 7, 9};

  (* Function to count primes in a specific residue class mod 10 *)
  piData[x_, r_] := Count[Select[Prime[Range[PrimePi[x]]], Mod[#, 10] == r &], _Integer];

  (* Logarithmic integral baseline for equidistribution *)
  liBase[x_] := N[LogIntegral[x]/4];

  (* Calculate deviations from the expected average density *)
  deviations = Table[
    {r, Table[{x, piData[x, r] - liBase[x]}, {x, xVals}]},
    {r, residues}
  ];

  (* Fetch imaginary parts of the first 15 nontrivial zeta zeros *)
  imZeros = Table[Im[ZetaZero[n]], {n, 1, 15}];

  (* Construct an oscillatory proxy: sqrt(x) * sum(cos(gamma * log(x))/gamma) *)
  model[x_] := Sqrt[x] * Total[Cos[imZeros * Log[x]] / imZeros];

  (* Generate visualization of prime deviations per residue class *)
  plot1 = ListLinePlot[
    deviations[[All, 2]],
    PlotLegends -> (StringForm["r = ``", #] & /@ residues),
    AxesLabel -> {"x", "Deviation from Li(x)/4"},
    PlotLabel -> "Prime Count Fluctuations Modulo 10",
    ImageSize -> Large
  ];

  (* Return the plot for comparison with spectral predictions *)
  Print["Spectral analysis complete for first ", limit, " integers."];
  plot1
]

The classification of composite numbers into terminal-digit families in arXiv:hal-01407334v1 provides a granular view of the sieve process. By shifting the perspective from the primes to the deterministic machinery of composite generation, we gain new tools to analyze the fluctuations of π(x). The most promising avenue for further research is the formal Fourier analysis of the union of these fifteen sets, which may eventually demonstrate that the frequencies of prime gaps are mathematically forced to the critical line.

References

• arXiv:hal-01407334v1: A New Sieve to Extract Primes and Its Connection to the Riemann Hypothesis.
• Davenport, H. (2000). Multiplicative Number Theory. Springer.
• Edwards, H. M. (1974). Riemann's Zeta Function. Academic Press.