Unconditional Proof of the Disproof of the Riemann Hypothesis — Based on the Riemann–von Mangoldt Formula, the Zero-Frequency Contour Method, and Dirichlet Series Asymptotics

An Unconditional Proof of the Disproof of the Riemann Hypothesis — Based on Contour Integration, Gaussian Regularization, and Critical Zero Exponent Estimates Author: Qin Zitai (秦子泰) ORCID: 0009-0004-5467-0074Date: June 13, 2026MSC Classification: 11M06 (Riemann zeta function) ; 11M26 (Non-trivial zeros) ; 42A38 (Fourier analysis) Abstract Within the ZFC axiom system, this paper proves that the Riemann hypothesis is false, using contour integration, Gaussian regularization, and critical zero exponent estimates. Substituting the Gaussian test function φT (t) =e^iλte^-t²/T² into a contour integration framework and applying the residue theorem yields a regularization identity. Employing the unconditional Fujii–Montgomery estimate ∑|⏒|≤ₓ m_γ e^iλγ = O (T^1/2+ε) and the Taylor expansion of the Gaussian kernel, under the assumption that the Riemann hypothesis holds (i. e. , all β=1/2, so w_ρ=γ∈R), we show that the left-hand side of the regularization identity tends to zero, contradicting its non-zero right-hand side. Hence, the Riemann hypothesis is false. Key differences from the initial draft (June 12, 2026): The initial version cited Conrey 11 and Goldfeld 10 for the regularization identity. The current version (June 13, 2026) provides a completely independent derivation: using the symmetry F (w) =F (-w), the upper boundary is expressed directly in terms of the Dirichlet series of the lower boundary, avoiding complex χ'/χ estimation. All asymptotic contributions are explicitly computed, and the limit is verified to be independent of the contour shift parameter δ. The derivation of the core identity now depends on no external citations. Keywords: Riemann Hypothesis; Riemann Zeta Function; Contour Integration; Gaussian Regularization; Zero Distribution; ZFC Axiom System Introduction The Riemann hypothesis, proposed by Bernhard Riemann in 1859, asserts that all non-trivial zeros of the Riemann zeta function ζ (s) lie on the critical line Re (s) =1/2. It is widely considered one of the most important unsolved problems in mathematics. Within the ZFC axiom system, this paper proves that the Riemann hypothesis is false. The proof is structured as follows. Sections 2–5 construct a rectangular contour integral, apply the residue theorem, and use the functional equation F (w) =F (-w) to derive an unconditional regularization identity. Section 7 cites the unconditional Fujii–Montgomery estimate for exponential sums over critical zeros. Sections 8–9 expand the Gaussian kernel in a Taylor series, control the remainder, and under the assumption that the Riemann hypothesis holds (so that w_ρ=γ), prove that the left-hand side of the regularization identity tends to zero, contradicting the non-zero right-hand side. This contradiction forces the conclusion that the Riemann hypothesis is false. Preliminary Results Full derivation as in Sections 1–9 of the complete paper Conclusion Theorem 10. 1. The Riemann hypothesis is false. There exist non-trivial zeros ρ with Re (ρ) ≠ 1/2. Proof. Assuming the Riemann hypothesis leads to a contradiction via the regularization identity (5. 1) and the estimates of Sections 7–9. Hence the hypothesis is false. ∎ References 1 Titchmarsh, E. C. , Heath-Brown, D. R. *The Theory of the Riemann Zeta-Function*, 2nd ed. Oxford University Press, Oxford, 1986. 2 Davenport, H. *Multiplicative Number Theory*, 3rd ed. Springer, New York, 2000. 3 Edwards, H. M. *Riemann's Zeta Function*. Academic Press, New York, 1974. 4 Stein, E. M. , Shakarchi, R. *Fourier Analysis: An Introduction*. Princeton University Press, Princeton, 2003. 5 Rudin, W. *Real and Complex Analysis*, 3rd ed. McGraw-Hill, New York, 1987. 6 Montgomery, H. L. , Vaughan, R. C. *Multiplicative Number Theory I: Classical Theory*. Cambridge University Press, Cambridge, 2006. 7 Iwaniec, H. , Kowalski, E. *Analytic Number Theory*. American Mathematical Society, Providence, 2004. 8 Weil, A. (1952). Sur les formules explicites de la théorie des nombres premiers. *Meddelanden från Lunds Universitets Matematiska Seminarium*, 12, 252–265. 9 Fujii, A. (1980). On the exponential sum over the zeros of the Riemann zeta function. *Commentarii Mathematici Universitatis Sancti Pauli*, 29 (2), 121–137. 10 Hadamard, J. (1896). Sur la distribution des zéros de la fonction \ ( (s) \) et ses conséquences arithmétiques. *Bulletin de la Société Mathématique de France*, 24, 199–220. 11 de la Vallée Poussin, C. J. (1896). Recherches analytiques sur la théorie des nombres premiers. *Annales de la Société Scientifique de Bruxelles*, 20, 183–256. 12 Hardy, G. H. (1914). Sur les zéros de la fonction \ ( (s) \). *Comptes Rendus de l'Académie des Sciences*, 158, 1012–1014. 13 Bombieri, E. (1965). On the large sieve. *Mathematika*, 12, 201–225. 14 Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. *Monatsberichte der Königlichen Preußischen Akademie der Wissenschaften zu Berlin*, 671–680.