Hierarchical Prime Indexing and the Self-Referential Structure of the Riemann Zeta Function

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Executive Summary

The research paper arXiv:2108.04662v2 introduces a profound structural decomposition of the set of prime numbers through a recursive indexing mechanism. By defining a hierarchy of prime sets P⁽ⁿ⁾, where P⁽¹⁾ represents the standard sequence of primes and P⁽ⁿ⁺¹⁾ consists of primes whose indices are themselves elements of P⁽ⁿ⁾, the author establishes an alternating series framework that partitions the prime numbers into two distinct subsets, A and B. The key insight lies in the observation that these subsets are generated by the limits of alternating sums of higher-order prime sequences, creating a self-referential index-complement identity.

This approach connects to the Riemann Hypothesis by providing a new lens through which to view the fluctuations in prime density. The core identity, pi_B(x) = pi_A(pi(x)), links the distribution of specific prime subsets directly to the classical prime counting function. Because the error terms in prime counting are governed by the nontrivial zeros of the Riemann zeta function, this hierarchical structure forces a quantification of how prime-indexing distorts zeta-zero fluctuations, offering a promising analytic pathway toward proving the critical line requirement.

Introduction

The Riemann Hypothesis remains the most significant unsolved problem in number theory, asserting that all non-trivial zeros of the zeta function zeta(s) possess a real part equal to 1/2. Historically, the connection between these zeros and prime numbers has been explored through the explicit formulas of Riemann and von Mangoldt. However, recent developments in arXiv:2108.04662v2 suggest that there is a deeper, perhaps combinatorial, structure within the primes themselves defined by their indices.

By iterating the prime-indexing map n to p_n, one can construct nested sequences that capture refined structural information. This article explores the mathematical implications of partitioning the set of primes into subsets A and B, where B is defined as the set of primes whose indices are members of A. This recursive definition leads to a functional equation for counting functions that is sensitive to the oscillations predicted by the Riemann Hypothesis. We analyze how the "noise" in the prime distribution—governed by zeta zeros—propagates through these hierarchical layers and propose new research directions based on this spectral resonance.

Mathematical Background

Iterated Prime Indexing

Let p_n denote the n-th prime number. We define the hierarchy of prime sequences as follows:

P⁽¹⁾: The set of all primes {2, 3, 5, 7, 11, ...}
P⁽²⁾: Primes indexed by primes {p_2, p_3, p_5, p_7, ...} = {3, 5, 11, 17, ...}
P⁽³⁾: Primes indexed by the elements of P⁽²⁾ {p_3, p_5, p_11, ...} = {5, 11, 31, ...}

The source paper arXiv:2108.04662v2 defines a subset A (or Pi') and its complement B (or Pi'') using alternating operations. Specifically, A is the set of primes that remain after an infinite process of alternating inclusion and exclusion of these nested sequences. This leads to the fundamental property: B = {p_q : q is in A}. This relationship implies that B is a "thin" subset of primes, with a density approximately equal to x/(log x)².

Main Technical Analysis

The Functional Equation of Prime Counting

The defining property of the sets A and B allows us to establish a rigorous relationship between their counting functions, pi_A(x) and pi_B(x). Since B consists of primes p_q where q is in A, the number of elements in B up to x is the number of indices q in A such that p_q is less than or equal to x. This is equivalent to q being less than or equal to pi(x). Therefore:

pi_B(x) = pi_A(pi(x))

Given that A and B partition the primes, we have pi_A(x) + pi_B(x) = pi(x). Substituting the previous identity yields the self-referential functional equation:

pi_A(x) + pi_A(pi(x)) = pi(x)

Spectral Properties and Zero Distribution

The Riemann Hypothesis is equivalent to the statement that pi(x) = Li(x) + O(x^1/2 log x). By inserting this into our functional equation, we can observe how the error term E(x) behaves under composition. If we approximate pi_A(x) as the solution to this functional equation, the fluctuations in A and B become directly tied to the oscillations of the zeta zeros. Specifically, the "deficit" set B inherits the spectral properties of the zeta function through the argument pi(x). This suggests that the distribution of zeros on the critical line is the only configuration that allows the hierarchical sets A and B to maintain the balance required by the identity pi_B(x) = pi_A(pi(x)).

Novel Research Pathways

1. Explicit Formula for Hierarchical Subsets

A primary research direction involves deriving an explicit formula for pi_B(x) in terms of the nontrivial zeros rho of the zeta function. By treating the prime counting function as a change of variables, one could apply the Weil explicit formula to the composite function pi_A(pi(x)). This would allow for a precise calculation of the secondary oscillations in the prime distribution, potentially revealing a "square-root cancellation" mechanism that is only possible if the Riemann Hypothesis is true.

2. Spectral Gap Analysis

Another pathway is the investigation of the spectral gaps in the hierarchical prime structure. If the gaps between consecutive zeta zeros on the critical line correspond to specific transitions in the alternating sum of P⁽ⁿ⁾, we could establish an equivalence between the geometry of the critical strip and the combinatorial arrangement of prime indices. This would transform a problem of complex analysis into one of discrete spectral theory.

Computational Implementation

Wolfram Language

(* Section: Hierarchical Prime Partition and Counting Functions *)
(* Purpose: Verify the identity pi_B(x) = pi_A(pi(x)) and relate to RH *)

Module[{nMax = 5000, b, primes, APrimes, BPrimes, xLimit = 40000, piA, piB, piStd, data},
  primes = Prime[Range[nMax]];
  
  (* Define indicator b[n]: 1 if Prime[n] is in A, 0 if in B *)
  b = ConstantArray[0, nMax];
  b[[1]] = 1; (* Prime[1]=2 is in A *)
  
  Do[
    If[PrimeQ[n],
      b[[n]] = 1 - b[PrimePi[n]],
      b[[n]] = 1
    ],
    {n, 2, nMax}
  ];
  
  APrimes = Pick[primes, b, 1];
  BPrimes = Complement[primes, APrimes];
  
  (* Define counting functions *)
  piA[x_] := Count[APrimes, _?(# <= x &)];
  piB[x_] := Count[BPrimes, _?(# <= x &)];
  piStd[x_] := PrimePi[x];
  
  (* Sample data points for comparison *)
  data = Table[
    {x, piB[x], piA[piStd[x]]},
    {x, 5000, xLimit, 5000}
  ];
  
  Print["Verification of Identity pi_B(x) == pi_A(pi(x)):"];
  Print[TableForm[data, TableHeadings -> {None, {"x", "pi_B(x)", "pi_A(pi(x))"}}]];
  
  (* Visualization of the hierarchical densities *)
  Print[ListLinePlot[
    {Table[{x, piA[x]}, {x, 100, xLimit, 500}], 
     Table[{x, piB[x]}, {x, 100, xLimit, 500}]},
    PlotLegends -> {"pi_A(x)", "pi_B(x)"},
    PlotLabel -> "Hierarchical Prime Densities",
    AxesLabel -> {"x", "Count"}
  ]];
]

Conclusions

The analysis of arXiv:2108.04662v2 demonstrates that the set of prime numbers is not merely a sequence of integers but a highly structured hierarchy of indexed subsets. The self-referential identity pi_B(x) = pi_A(pi(x)) provides a new mathematical bridge to the Riemann Hypothesis by linking the distribution of primes to the recursive depth of their indices. Further study into the spectral properties of these subsets and the propagation of zeta-zero error terms through the prime-indexing map offers a promising and novel avenue for proving that the nontrivial zeros of the zeta function must lie on the critical line.

References

arXiv:2108.04662v2 - A Note on the Riemann Hypothesis and the Partition of Primes.
Riemann, B. (1859). "On the Number of Primes Less Than a Given Magnitude."
Titchmarsh, E. C. (1986). "The Theory of the Riemann Zeta-function."