Riemann HypothesisDirect VerificationRiemann-Siegel

Riemann Hypothesis Verification

Direct computation of zeros of ζ(s) on the critical line Re(s) = ½ using the Riemann-Siegel Z(t) function. Every sign change of Z(t) corresponds to a verified zero of the Riemann zeta function on the critical line.

How this works: The Riemann-Siegel formula computes Z(t) = e^iθ(t)ζ(½ + it), which is real-valued. Sign changes of Z(t) prove that ζ(½ + it) = 0, directly confirming RH for each zero found. Over 10¹³ zeros have been verified this way.

Search range

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Resolution (sample points)

First known zero: t ≈ 14.1347. Higher t values require more computation. Recommended: t < 1000, resolution 5000–50000.