# What is the formula Mathematica uses for ZetaZero?

What is the formula/algorithm Mathematica uses for the ZetaZero command?

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Try contacting wolfram support and ask them. – Szabolcs May 22 '14 at 23:45

It appears to look for Gram points, which can be approximated by

$g_n \approx 2\pi\exp\left(1 + W\left(\frac{8n+1}{8e}\right)\right),$

where $W$ is the Lambert W function and then numerical root finding can zero into $g_n$.

Gram points can help find zeta zeros, which is touched on here.

Edit: I found this by traversing through DownValues, so it may be way more complicated than this.

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It seems really close, but I need something far more precise. I've been working on the Riemann Hypothesis (yes, I'm well aware that I'll probably never solve it.) Thanks for the help. – Paige May 22 '14 at 21:14
Well I will say that ZetaZero explicitly assumes RH by looking for Gram points, i.e. it only looks on the critical line. – Chip Hurst May 23 '14 at 3:57
Also for what it's worth you can read about using Gram points to find zeros starting on page 119 here: books.google.com/… – Chip Hurst May 23 '14 at 4:37
To add to this: after using the Gram points to bracket a root, the Riemann-Siegel Z function $Z(t)$ is then used for the actual root finding, since it has been designed to be zero whenever $\zeta\left(\frac12+it\right)$ is zero. – J. M. Jun 11 '15 at 10:35