Argument principle states:
If $f$ is a meromorphic function inside and on some closed contour $C$, and $f$ has no zeros or poles on $C$, then $$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=Z-P$$ where $Z$ and $P$ denote the number of zeros and poles in $C$ (counting multiplicity).
The central part of numerically verifying the Riemann hypothesis is using the argument principle to count the number of zeros of the Riemann zeta function in a given region.
But there is one problem: When choosing the contour, how do we know that $\zeta$ has no zeros on the contour? We don't know whether a region is zero-free in advance, so how should we choose the contour?
My idea was that the imaginary parts of the zeros are all irrational, so we just choose a rational number, but it turns out that we actually don't know if the imaginary parts are irrational.