Download Full Article
This article is available as a downloadable PDF with complete code listings and syntax highlighting.
Executive Summary
The research paper arXiv:2210.08614v1 introduces a rigorous combinatorial identity for the prime pair counting function, often denoted as pi(2)(N). By partitioning the sum of prime counting function evaluations at the square root of N, the paper establishes an exact relationship: the difference between the sum of pi(N/p) for primes below the square root and the sum for primes above it equals the square of the prime counting function at the square root. This hyperbolic symmetry provides a novel framework for analyzing the distribution of primes. Because the Riemann Hypothesis (RH) is equivalent to tight bounds on the error term of the prime counting function, this identity acts as a diagnostic tool. It translates the analytical behavior of zeta function zeros into a discrete combinatorial constraint, offering a promising new pathway to investigate the critical line through the lens of semiprime variance.
Introduction
The distribution of prime numbers is governed by the Riemann zeta function, where the location of its non-trivial zeros dictates the precision of the Prime Number Theorem. While the counting of individual primes is well-studied, the distribution of semiprimes (numbers formed by the product of exactly two primes) offers a higher-order perspective on these arithmetic fluctuations. Traditionally, semiprime counting has been approached through asymptotic methods, such as those established by Edmund Landau.
However, the source paper arXiv:2210.08614v1 moves beyond asymptotics to provide an exact identity. This identity relates the local density of primes (up to the square root of N) to the global density (up to N/2). The significance of this result lies in its structure; it creates a balanced equation where any deviation from the predicted prime density—such as those caused by zeros off the critical line—would manifest as a measurable discrepancy in the semiprime count. This article provides a technical analysis of these structures and proposes how they might be used to verify the Riemann Hypothesis computationally and theoretically.
Mathematical Background
To analyze the findings of arXiv:2210.08614v1, we define the prime counting function pi(x) as the number of primes less than or equal to x. The paper focuses on the enumeration of ordered pairs (p, q) such that their product is less than or equal to N. The central discovery is a formula for the number of such pairs that avoids recursive complexity.
The primary identity is formulated as follows: the sum of floor(pi(N/p_k)) for all primes p_k from k = 1 to pi(sqrt(N)), when contrasted against the sum for primes from the square root up to N/2, yields exactly pi(sqrt(N)) squared. Mathematically, this is expressed as:
- Sum A: The total counts of primes q such that q is less than or equal to N/p, where p is a prime less than or equal to the square root of N.
- Sum B: The total counts of primes q such that q is less than or equal to N/p, where p is a prime between the square root of N and N/2.
- The Identity: Sum A - Sum B = (pi(sqrt(N)))2.
This result is grounded in the Dirichlet hyperbola method, but it is unique because it uses the indicator function of primality as its weight. It establishes a rigid "skeleton" for prime distribution that must be maintained for the identity to hold for every integer N.
Main Technical Analysis
Hyperbolic Symmetry and the Critical Threshold
The appearance of the square root of N as a threshold is mathematically significant. In the study of L-functions, this threshold often represents the point of symmetry in functional equations. In the context of arXiv:2210.08614v1, the square root of N serves as the pivot point for a bilinear prime correlation. Sum A counts pairs where at least one prime is small, while Sum B counts pairs where both primes are of intermediate size.
Error Propagation and Zeta Zeros
Under the Riemann Hypothesis, the error term E(x) = pi(x) - Li(x) is bounded by O(x1/2 log x). When we substitute this into the identity from the source paper, we find that the total error in the semiprime count is an aggregate of these individual errors across different scales. Specifically, the sum over E(N/p_k) must satisfy the combinatorial constraint that the final result remains an integer (pi(sqrt(N)))2.
If a zero of the zeta function existed with a real part greater than 1/2, it would introduce large-scale oscillations into the prime counting function. These oscillations would propagate through the sums in the source paper's formula. Because the identity is exact, these oscillations would have to cancel out perfectly across the range [2, N/2] to avoid violating the identity. This suggests that the distribution of primes is even more constrained than previously thought, as it must satisfy these second-order combinatorial symmetries.
Novel Research Pathways
1. The Hyperbola Cancellation Criterion
Researchers can define a discrepancy function R(N) as the difference between the exact identity in arXiv:2210.08614v1 and its smoothed approximation using the logarithmic integral Li(x). A promising research direction is to prove that if R(N) is bounded by a square-root growth factor, then the Riemann Hypothesis must hold. This would transform the analytical problem into a study of cancellation in correlated prime sums.
2. Moment Bounds and Zero-Free Regions
By applying the identity to Dirichlet L-functions, one could investigate the Generalized Riemann Hypothesis. If the semiprime identity holds for primes in arithmetic progressions, it places severe limits on the existence of Siegel zeros. Analyzing the second moment of the zeta function through these partitioned sums could yield improved constants for the zero-free region of the zeta function.
Computational Implementation
The following Wolfram Language code verifies the fundamental identity from arXiv:2210.08614v1 and visualizes the fluctuations of the semiprime counting function relative to smooth approximations.
(* Section: Verification of Semiprime Identity and Fluctuation Analysis *)
(* Purpose: Verify the identity from arXiv:2210.08614v1 and check residuals *)
Module[{nMax = 1000, testNs, results, fluctuationData},
(* Define the sum-based identity terms *)
S1[n_] := Total[Table[PrimePi[n/Prime[k]], {k, 1, PrimePi[Sqrt[n]]}]];
S2[n_] := Total[Table[PrimePi[n/Prime[k]], {k, PrimePi[Sqrt[n]] + 1, PrimePi[n/2]}]];
(* Verify identity: S1 - S2 should equal PrimePi[Sqrt[n]]^2 *)
VerifyIdentity[n_] := S1[n] - S2[n] - PrimePi[Sqrt[n]]^2;
(* Run verification for a range of N *)
testNs = Range[100, 1000, 50];
results = Table[{n, VerifyIdentity[n]}, {n, testNs}];
Print["Residuals of the Identity (Should all be 0):"];
Print[TableForm[results, TableHeadings -> {"N", "Residual"}]];
(* Fluctuation Analysis: Compare exact count to Li-based model *)
fluctuationData = Table[
{n, (S1[n] + S2[n]) - LogIntegral[n] * Log[Log[n]]},
{n, 100, 2000, 20}
];
ListLinePlot[fluctuationData,
PlotLabel -> "Normalized Fluctuation of Semiprime Sums",
AxesLabel -> {"N", "Deviation"},
PlotStyle -> Red]
]Conclusions
The identities established in arXiv:2210.08614v1 provide a rigorous bridge between the combinatorial counting of prime pairs and the analytic properties of the Riemann zeta function. By proving that the distribution of primes must satisfy a specific hyperbolic symmetry, the paper offers a new set of constraints for the location of zeta zeros. The most promising next step is the formalization of the "Hyperbola Cancellation Criterion," which could potentially lead to a proof that the observed stability of semiprime counts necessitates the truth of the Riemann Hypothesis. This approach moves the field closer to an elementary yet profound understanding of the critical line.
References
- Source Paper: arXiv:2210.08614v1
- Related Context: arXiv:1402.3612v2
- Classic Reference: Riemann, B. (1859). "On the Number of Primes Less Than a Given Quantity."