A “The Files are in the Computer” moment

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 ${U}$ be any compact set of complex numbers whose real parts ${x}$ satisfy ${\frac{1}{2} < x < 1}$ . Because ${U}$ is closed and bounded, there exists a fixed ${r < \frac{1}{4}}$ such that ${\frac{3}{4} - r \leq x \leq \frac{3}{4} + r}$ for all ${z \in U}$ with real part ${x}$ . The ${\frac{3}{4} - r}$ part can be as close as desired to the critical line ${x = \frac{1}{2}}$ , but it must stay some fixed distance away.

We also need ${U}$ to have no “holes,” i.e., to be homeomorphic to a closed disk of radius ${r}$ . Often ${U}$ is simply taken to be the disk of radius ${r}$ centered on the ${x}$ -axis at ${\frac{3}{4}}$ , but we’ll also think of a square grid. Then Voronin proved:

Theorem 1Given any analytic function ${f(z)}$ that is non-vanishing on ${U}$ , and any ${\epsilon>0}$ , we can find some real value ${t}$ such that for all ${z \in U}$ ,

$\displaystyle |\zeta(z+it)-f(z)| < \epsilon.$