Combinatorial Prime Factorization Slots and the Critical Line: A New Approach to the Riemann Hypothesis

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Introduction

The Riemann Hypothesis remains the most profound challenge in analytic number theory, asserting that all non-trivial zeros of the zeta function ζ(s) lie on the critical line Re(s) = 1/2. Traditionally, investigations into this hypothesis have been dominated by complex analysis and spectral theory. However, the source paper arXiv:hal-01176804 introduces a compelling combinatorial bridge. By reframing the set of natural numbers as a result of combinatorial distributions of N primes into q factor slots, the paper provides a framework for evaluating the density of integers with specific factorization parities.

This approach is significant because it reframes the set of natural numbers as a limit of finite combinatorial structures. By defining Q_q^N as the number of distinct integers that can be formed using exactly q prime factors from a set of N primes, the paper establishes a direct link between binomial coefficients and the cardinality of subsets of the natural numbers. This analysis explores how the asymptotic behavior of these sequences exhibits oscillatory patterns that mirror the non-trivial zeros of the Riemann zeta function.

The motivation for this study lies in the parity of prime factors, often represented by the Liouville function λ(n). If the distribution of integers with an even versus an odd number of prime factors is sufficiently balanced, the resulting cancellations in the Dirichlet series ensure that the zeta function has no zeros to the right of the critical line. The source paper provides the combinatorial machinery to bound these fluctuations using the properties of binomial coefficients and hyperbolic functions.

Mathematical Background

The foundational structures in arXiv:hal-01176804 center on representing natural numbers through combinatorial arrangements of prime factors in "slots." For a natural number possessing q prime factors, counting multiplicity, the paper introduces combinatorial quantities that enumerate different factorization patterns.

Key Combinatorial Objects

Factorization Count Q_q^N: Defined as C(N+q-1, q), this represents the number of ways to choose q elements from a set of N elements with replacement. This corresponds to the number of integers that can be formed using q prime factors from a set of N available primes.
Parity Sums s_k^N: The paper defines s_k^N = Q_2k^N + Q_2k-1^N. This grouping allows for the analysis of the growth of the total number of integers as the number of factors increases.
Parity Differences D_k^N: Conversely, the difference D_k^N = Q_2k^N - Q_2k-1^N is crucial for understanding the imbalance between even and odd prime factor counts.

Through detailed algebraic manipulation, the source paper establishes that s_k^N = (N+4k-1)/(2k)! multiplied by a product of terms from (N-1)+1 to (N-1)+2k-1. This expression reveals the relationship between the scale N, the slot number k, and the resulting combinatorial weight. The paper confirms that the total number of natural numbers is the limit of these combinatorial sums as N approaches infinity, establishing a direct connection between combinatorial enumeration and cardinality.

Symmetric Polynomials and Parity-Weighted Euler Products

The transition from the discrete combinatorial counts Q_q^N to the continuous properties of the zeta function occurs through the generating function of the Euler product. The fundamental limitation of raw counts is that they treat each prime as a symbol rather than a numerical value. To connect arXiv:hal-01176804 to the Riemann Hypothesis, we must pass from slot-counting to analytically meaningful weightings.

Mathematically, Q_q^N is the coefficient of x^q in (1 - x)^-N. The appropriate weighted replacement is the complete homogeneous symmetric polynomial h_q(x₁, ..., x_N). If we set x_j = p_j^-s, where p_j is the j-th prime, the combinatorial object becomes an analytic one. The parity-weighted combination is particularly revealing: the difference between the even part and the odd part converges to the Dirichlet series of the Liouville function, ζ(2s)/ζ(s). This is the first genuinely adjacent bridge: zeros of ζ(s) become poles of this ratio, and analytic control of the sum as N increases is sensitive to the zero distribution.

A technically notable step in the source paper is bounding the sum of s_k^N by a series with (2k)! in the denominator, leading to a hyperbolic cosine majorant: the sum of s_k^N is less than cosh((1 + 1/α)N) - 1. While this is a combinatorial estimate, it suggests a general method: transforming discrete growth into an entire function bound by forcing factorial denominators. This resembles the structure of bounding exponential expansions of Dirichlet polynomials, which is a key technique in controlling the growth of the zeta function within the critical strip.

Novel Research Pathways

Pathway 1: Spectral Analysis of Combinatorial Zeta Functions

The combinatorial structures in arXiv:hal-01176804 naturally define a family of zeta-like functions through the generating series of s_k^N. We propose investigating the spectral properties of a combinatorial zeta function where the coefficients are derived from the limit of binomial counts. The research methodology involves establishing the analytic continuation of this function through the functional equation derived from combinatorial symmetries. The expected outcome is a demonstration that the non-trivial zeros of such a function must lie on the line Re(s) = 1/2, providing a combinatorial analog to the Riemann Hypothesis.

Pathway 2: Prime Gap Analysis through Difference Sequences

The sequence D_k^N encodes information about the asymmetry between even and odd factorization patterns, which directly relates to prime gap distributions. We propose linking combinatorial differences to the gaps between consecutive primes. If the growth of D_k^N can be rigorously bounded in terms of N, it may imply square-root cancellation in the summatory Liouville function. This would provide a combinatorial proof that consecutive prime gaps satisfy the bounds required by the Riemann Hypothesis.

Pathway 3: Truncated Euler Products and Zero-Distance Penalties

One can view the s_k^N estimates as a majorant for controlling expansions where the true coefficients are weighted by primes. We propose a methodology to produce a uniform majorant for truncated Euler products that resembles the paper's cosh bound, but incorporates an explicit "zero penalty term." By investigating how the breakdown of such uniformity correlates with proximity to zeta zeros, we can identify the specific combinatorial obstructions to proving the hypothesis on the critical line.

Computational Implementation

The following Wolfram Language code demonstrates the combinatorial counts and the parity differences discussed in the source paper, allowing for a visual verification of the growth rates and their relation to the non-trivial zeros of the zeta function.

(* Section: Combinatorial Parity and Zeta Zero Correlation *)
(* Purpose: This code computes the combinatorial sequences s_k^N and D_k^N, 
   constructs a truncated Euler product, and visualizes the relationship 
   between combinatorial density and the zeros of the zeta function. *)

ClearAll[qSlot, sSlot, dSlot, truncEuler, zeros, alphaBound];

(* Define the fundamental combinatorial count from hal-01176804 *)
qSlot[n_Integer, q_Integer] := Binomial[n + q - 1, q];

(* Define the parity sum and difference sequences *)
sSlot[n_Integer, k_Integer] := qSlot[n, 2*k] + qSlot[n, 2*k - 1];
dSlot[n_Integer, k_Integer] := qSlot[n, 2*k] - qSlot[n, 2*k - 1];

(* Truncated Euler product over the first N primes *)
truncEuler[s_, n_Integer] := Product[1/(1 - Prime[j]^-s), {j, 1, n}];

(* Hyperbolic bound check as described in the source paper *)
alphaBound[n_, alpha_] := Cosh[(1 + 1/alpha)*n] - 1;

(* Extract first 15 non-trivial zeros for comparison *)
zeros = Table[Im[ZetaZero[k]], {k, 1, 15}];

(* Generate visual comparison on the critical line Re(s) = 1/2 *)
Module[{nPrimes = 100, tMax = 60},
  plot1 = Plot[Log[Abs[truncEuler[1/2 + I*t, nPrimes]]], {t, 0, tMax},
    PlotRange -> All,
    Frame -> True,
    FrameLabel -> {"t = Im(s)", "Log|E_N(s)|"},
    PlotStyle -> Blue,
    PlotLabel -> "Truncated Euler Product vs. Zeta Zeros"];

  (* Mark the locations of actual Zeta zeros *)
  plot2 = Graphics[{Red, Opacity[0.5], 
    Table[Line[{{z, -10}, {z, 10}}], {z, zeros}]}];

  (* Display the combined plot *)
  Print[Show[plot1, plot2]];
];

(* Output a table of combinatorial values for N=5 *)
Table[{k, sSlot[5, k], dSlot[5, k], N[sSlot[5, k]/alphaBound[5, 2]]}, 
  {k, 1, 8}] // TableForm

Conclusions

The combinatorial framework presented in arXiv:hal-01176804 provides a robust foundation for analyzing the structure of natural numbers through prime factorization slots. By deriving explicit bounds for the sums and differences of integers with specific factor parities, the paper offers a new lens through which the Riemann Hypothesis can be viewed. The transition from finite prime bases to the infinite limit successfully recovers the totality of the natural numbers, suggesting that these combinatorial inequalities are fundamental properties of the integers.

The most promising avenue for further research lies in the integration of these combinatorial counts with the actual magnitudes of the primes. A hybrid model that combines the slot-based counting with analytic estimates of prime density could potentially yield a proof of the square-root growth of the summatory Liouville function, thereby resolving the Riemann Hypothesis.

References

arXiv:hal-01176804 - The totality of the natural numbers and the Riemann Hypothesis.
arXiv:hal-01220071 - Combinatorial approach to the distribution of prime factors.
Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function. Oxford University Press.