Are the $\Im(s)$ for zeros of the Riemann zeta function isolated values?

Question

I am interested in doing a video about the geometry of the Riemann zeta function, which is to include diagrams of the critical strip. In all of the diagrams I've seen, zeros are isolated points. I'm not sure that has been proven.

An earlier question referred to the article https://phys.org/news/2017-04-insight-math-million-dollar-problem-riemann.html which states without reference that "One of the most helpful clues for proving the Riemann hypothesis has come from function theory, which reveals that the values of the imaginary part, t, at which the function vanishes are discrete numbers." One of the answers indicated that "discrete" means "isolated."

Can anyone provide a reference for that claim?

Should I draw possible zeros (symmetrically across $\Re(s) = 1/2$) as points, line segments of constant $\Re(s)$, line segments of constant $\Im(s)$, and disks?

Answer 1

Let $a+ib$ be a non-trivial zero of $\zeta$, say with $b>0$. Then any zero with imaginary part between $b/2$ and $2b$ lies in the box $$B=\{x+iy:0\le x\le1,b/2\le y\le 2b\}.$$ The box $B$ is compact, and $\zeta$ is holomorphic on an open set containing $B$. As the zeros of a holomorphic function are isolated, then $\zeta$ has finitely many zeros on $B$. Therefore there are only finitely many numbers in the interval $[b/2,2b]$ which are imaginary parts of zeros, and so there's an open interval centred at $b$ in which $b$ is the only possible imaginary part of a zero of $\zeta$.