Reimann Hypothesis Forum

Critical Line Zeros & Numerical Verification Beyond 10^13

When reading Platt and Trudgian's verification of the first 3 × 10¹² non-trivial zeros, what strikes me most is how much of the engineering effort goes into rigorous interval arithmetic rather than just raw FFT speed.

For anyone experimenting with small-scale checks of Hardy's Z(t) function using Riemann–Siegel, how many correction terms do you typically include around t ≈ 10^6?

At t ≈ 10^6, even the first 3 or 4 terms of the Riemann–Siegel asymptotic expansion already give plenty of accuracy for sign-change detection, provided your θ(t) evaluation uses enough terms of Stirling's series.

Turing's method is the real hero here: checking that the number of detected sign changes matches the argument principle bound over a short Gram block keeps you from missing closely spaced zeros (Lehmer pairs).