What question did this study set out to answer?

The central aim is to prove the Riemann Hypothesis through a series of iterative steps involving complex rewriting.

March 8, 2026Open Access

A Fixed-Point Reformulation of the Riemann Hypothesis: Iterative Complex Rewriting of n and Convergence on the Critical Line

Key Points

The central aim is to prove the Riemann Hypothesis through a series of iterative steps involving complex rewriting.
Utilized a complex rewriting lemma n^(s_n) = n
Applied Euler's reflection formula to show F(s)·F(1-s) = 1
Demonstrated symmetry on the critical line with |F(s)| = 1 iff sigma = 1/2
Differentiated the function xi(s) to establish properties of its derivatives at critical points
Derived conditions linking the zeta function and G function at corresponding critical values.
Proved that |F(s_0)| = 1 on the critical line, reinforcing sigma = 1/2
Showed that no assumptions about sigma were needed throughout the proof
Established a rigorous link between derivatives of xi and zeta functions at critical points.

Abstract

Version 7. 0 — Complete proof. We establish the Riemann Hypothesis via a chain of five steps: (1) Complex rewriting lemma n^ (sₙ) = n; (2) F (s) ·F (1-s) = 1 via Euler's reflection formula; (3) |F (s) | = 1 iff sigma = 1/2, from the symmetry 1-s = conjugate (s) on the critical line; (4) From xi (s) = xi (1-s), differentiating k times: |xi^ (k) (s₀) | = |xi^ (k) (1-s₀) | without any assumption on sigma. At a zero of order k, lower terms vanish and xi^ (k) (s₀) = G (s₀) ·zeta^ (k) (s₀), giving |zeta^ (k) (s₀) |/|zeta^ (k) (1-s₀) | = |G (1-s₀) |/|G (s₀) | = |F (s₀) |; (5) Therefore |F (s₀) | = 1, hence sigma = 1/2. No assumption on sigma is made at any step. The argument uses only xi (s) = xi (1-s) (Riemann) and Euler's reflection formula. Independent verification by the mathematical community is invited. ORCID: 0009-0007-4590-9874

A Fixed-Point Reformulation of the Riemann Hypothesis: Iterative Complex Rewriting of n and Convergence on the Critical Line

Key Points

Abstract

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