We establish a direct connection between knot theory and the Riemann Hypothesis. We prove that any complex matrix S satisfying S + ST = I has all eigenvalues with real part 1/2. We then construct an explicit sequence of such Seifert matrices SN from the prime numbers, where SN = 1/2 IN + i/2 KN with KNi, j = tan (π ln pⱼ) for i < j. The Alexander polynomials det (tSN - SNT) of this sequence are shown to converge to 1/ζ (s) after the change of variable t = exp (2π (s-1/2) ). This construction provides a topological mechanism for the Riemann Hypothesis: the non-trivial zeros of ζ (s) correspond to the eigenvalues of the limiting matrix, which by the No-Vacuum Principle must lie on the critical line Re (s) = 1/2. Numerical verification for N=59 shows agreement with the first Riemann zeros to within 0. 02. This work reduces the Riemann Hypothesis to a problem in linear algebra and knot theory.
Abdelilah AHMOURI (Thu,) studied this question.