GLL, The Amazing Zeta Code, here. It’s like Zoolander, the analytic functions are in the Zeta function! Why am I learning about this now for the first time? Look at the link to the number theory and physics archive as well, here. Best thing you will browse for a week.
The Amazing Property
Let be any compact set of complex numbers whose real parts satisfy . Because is closed and bounded, there exists a fixed such that for all with real part . The part can be as close as desired to the critical line , but it must stay some fixed distance away.
We also need to have no “holes,” i.e., to be homeomorphic to a closed disk of radius . Often is simply taken to be the disk of radius centered on the -axis at , but we’ll also think of a square grid. Then Voronin proved:
Theorem 1Given any analytic function that is non-vanishing on , and any , we can find some real value such that for all ,