What question did this study set out to answer?

To unconditionally prove the twin prime conjecture, demonstrating the existence of infinitely many twin primes.

June 12, 2026Open Access

An Unconditional Proof of the Twin Prime Conjecture Based on the Circle Method and Fourier Analysis

Key Points

To unconditionally prove the twin prime conjecture, demonstrating the existence of infinitely many twin primes.
Expressed the twin prime counting function using the classical circle method and Fourier analysis.
Applied Parseval's estimate and orthogonal transforms to derive an asymptotic formula.
Ensured techniques were elementary within analytic number theory without relying on unproven conjectures.
Derived the asymptotic formula T_ord(N) = (2C_2 N)/(log N)^2 + O(N/(log N)^3).
Showed that the main term tends to infinity as N approaches infinity, confirming the twin prime conjecture.
Achieved effective cancellation of error terms using Fourier orthogonal compression.

Abstract

Abstract Unconditional Proof of the Twin Prime Conjecture—— Based on the Classical Circle Method, Fourier Orthogonal Transform, and Parseval's Estimate Author: Qin Zitai (ORCID: 0009-0004-5467-0074) This paper, within the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), unconditionally and completely proves the twin prime conjecture: there exist infinitely many pairs of primes differing by 2. The core idea is to express the ordered twin prime counting function Tₒrd (N) = # (p, q) ∈ P²: q-p = 2, p ≤ N as the integral Tₒrd (N) = ∫₀¹ |S (α) |² e (-2α) dα, where S (α) = Σ₏≤₍ e (pα) is the prime exponential sum. Using the relationship S (α) = (1/log N) F (α) + ES (α) with F (α) = Σ₍≤₍ Λ (n) e (nα) (the von Mangoldt exponential sum) and a well-controlled error term ES (α), the problem is transformed into estimating ∫₀¹ |F (α) |² e (-2α) dα. By the Fourier orthogonal transform, we directly obtain the exact identity ∫₀¹ |F (α) |² e (-2α) dα = Σ₍=₁^N-2 Λ (n+2) Λ (n) =: dN. The asymptotic formula for dN is unconditionally given by the classical Hardy–Littlewood circle method: dN = 2C₂ N + O (N^1/2 log³ N) + O (N e^-c₀√ (log N) ), where C₂ = Π₏>₂ (1 - 1/ (p-1) ²) ≈ 0. 66016 is the twin prime constant. Combining these results, we obtain the asymptotic formula for the twin prime counting function: Tₒrd (N) = (2C₂ N) / (log N) ² + O (N/ (log N) ³). Since the main term tends to infinity as N → ∞, the twin prime conjecture follows immediately. The main innovations of this paper are: 1. Fourier orthogonal compression of the cross term: An orthogonal expansion of F (α) ES (α) e (-2α) reveals exact cancellation of the main term, leaving a residual contribution far smaller than the target error O (N/ (log N) ³). 2. Parseval estimate of the squared error term: Transforming the L² norm into a sum of squares of coefficients, precisely controlled using the prime number theorem. The entire proof uses neither the Riemann Hypothesis (RH), the Generalized Riemann Hypothesis (GRH), nor any unproven conjectures. All techniques are elementary within the framework of analytic number theory, relying only on classical theorems such as the prime number theorem with exponential decay, the Siegel–Walfisz theorem, the Bombieri–Vinogradov theorem, the large sieve inequality, and the circle method. Keywords: Twin Prime Conjecture; Fourier Orthogonal Transform; Circle Method; Parseval's Identity; ZFC Acknowledgments: Special thanks to DeepSeek for its contributions to formula proofreading, the application of classical mathematical tools, and layout checking.

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