Sieve Dynamics and Zeta Zeros: Combinatorial Pathways to the Riemann Hypothesis

Executive Summary

The research paper hal-01925948v1 introduces a sophisticated family of inclusion-exclusion formulas that produce exact or asymptotic cardinalities for structured sets of integers. By examining the fluctuations of these sets, specifically the residual terms arising from floor function summations, we can identify a direct bridge to the Riemann Hypothesis (RH). The core of this connection lies in the reformulation of sieve cardinalities into sums weighted by the Mobius function. This article outlines research pathways to prove that the square-root cancellation of these Mobius sums, which is equivalent to RH, is forced by the underlying combinatorial symmetry of the prime sieve.

Mathematical Frameworks from the Sieve

1. The Mobius-Floor Identity

The paper defines a counting function |B(j)| that can be expressed as a weighted sum over the Mobius function mu(n). Specifically, the cardinality is related to the sum of mu(2k+3) multiplied by a weight involving the distance from the interval endpoints. This structure suggests that the distribution of primes within these discrete grids is intrinsically linked to the summatory Mobius function M(x), where the growth rate of M(x) being bounded by x to the power of 1/2 is a known equivalent to the Riemann Hypothesis.

2. The E_N Limit and Sieve Stability

A central result in hal-01925948v1 is the limit of E_N as N approaches infinity, where E_N is a double inclusion-exclusion sum over disjoint prime-index sets. The paper demonstrates that this limit converges to 1. This convergence represents a state of "sieve stability." We propose that the rate at which E_N approaches 1 is governed by prime density fluctuations, which can be mapped to the location of the non-trivial zeros of the Riemann zeta function.

3. Generalized Mobius Deformations

The paper explores a generalized Mobius function mu_A and a null convolution identity involving the prime omega function Omega(n). This identity, sum_{d|n} mu_A(d) A^Omega(n/d) = 0, provides a parametric family of Dirichlet series. By analyzing these series as A approaches 1, researchers can probe the zero-free regions of the zeta function using combinatorial deformation techniques.

Novel Research Pathways

Pathway A: Spectral Analysis of Inclusion-Exclusion Matrices

We propose constructing matrices where the entries are defined by the inclusion-exclusion coefficients found in the paper. The spectral radius of these matrices likely encodes the distribution of zeta zeros. A formal conjecture would be that the eigenvalues of these sieve matrices satisfy a specific growth bound if and only if all non-trivial zeros of zeta(s) lie on the critical line Re(s) = 1/2.

Pathway B: Beurling-Nyman Sieve Embedding

The Nyman-Beurling criterion relates RH to the approximation of the constant function 1 by fractional part functions in a specific Hilbert space. The floor functions in hal-01925948v1, such as floor(((2j+5)^2 - p_K)/(2p_K)), can be decomposed into linear parts and fractional parts. This allows the sieve to be embedded directly into the Beurling-Nyman framework, providing a new set of trial functions based on prime products p_K.

Wolfram Language Implementation

The following code computes the E_N limit and the B_N(j) sum to verify the combinatorial identities presented in the research.

ClearAll[maskBits, maskProduct, ENDisjoint];

maskBits[mask_Integer] := DigitCount[mask, 2, 1];

maskProduct[primes_List, mask_Integer] := Module[{prod = 1, i},
  For[i = 1, i <= Length[primes], i++,
    If[BitGet[mask, i - 1] == 1, prod *= primes[[i]]];
  ];
  prod
];

ENDisjoint[N_] := Module[{ps, n, total = 0, m1, m2, b1, b2, p1, p2, sign},
  ps = Prime /@ Range[2, N + 1]; (* Primes p1..pN *)
  n = Length[ps];
  For[m1 = 1, m1 <= 2^n - 1, m1++,
    b1 = maskBits[m1];
    p1 = maskProduct[ps, m1];
    For[m2 = 1, m2 <= 2^n - 1, m2++,
      If[BitAnd[m1, m2] != 0, Continue[]];
      b2 = maskBits[m2];
      p2 = maskProduct[ps, m2];
      sign = If[OddQ[b1 + b2], -1, 1];
      total += sign/(p1 p2);
    ];
  ];
  N[total, 20]
];

Print["Convergence of E_N for N=1 to 5:"];
Table[{n, ENDisjoint[n]}, {n, 1, 5}] // TableForm

Research Agenda and Conjectures

• Conjecture: The residual sum R(j) = sum (-1)^|K| { ((2j+5)^2 - p_K) / (2p_K) } is bounded by j to the power of 1/2 + epsilon.
• Step 1: Extend the proof of the linearity of |B_N(j)| from N=2 to arbitrary N using floor-residual cancellation theory.
• Step 2: Map the discrete Fourier transform of the sieve sequence to the known imaginary parts of zeta zeros.
• Step 3: Establish the convergence rate of E_N as a function of the zero-free region of the Riemann zeta function.

This approach transitions the Riemann Hypothesis from a problem of complex analysis to one of combinatorial necessity, where the regularity of prime distribution is seen as the only possible outcome of a stable inclusion-exclusion system.