Riemann Rambling On

Amongst physicists, Riemann is best known for the concept of metric. But his one paper on number theory, where he defined the zeta function, was not the only work he did on the subject. Riemann actually computed some zeroes of $\zeta (s)$ himself. This was unknown for about 60 years, until Siegel went through some work of Riemann's and found a key to computing zeroes simply.

The Z function is defined by

$Z (t) = e^{i \theta (t)} \zeta (\frac{1}{2} + it)$

where

$\theta (t) = \textrm{arg} (\Gamma (\frac{2 i t + 1}{4})) - \frac{\textrm{log} \pi}{2} t$

The real zeroes of $Z (t)$ are the zeroes of $\zeta (s)$ on the critical line $s = \frac{1}{2} + it$. Positive real values of $t$ for which the $\zeta$ function is real are known as Gram points. By looking for pairs of Gram points, one can narrow down an interval where a zero of $\zeta$ must lie.

It is still rather impressive that Riemann managed to compute zeroes this way, with pen and paper, needing numbers such as the square root of 2 to something like 30 decimal places.