Zeros off the critical line, but extremely close to it

Question

I read the book Prime Obsession, and it seems that in order to determine if all the zeros in a range in the critical strip, one need not actually find and calculate each zero, but rather calculate some kind of contour integral using numerical methods that encompasses that range of interest. What if the there are zeros off the critical line, but extremely close to it, let's say by < 1/2 +/- 10^-10000? It seems to me that numerical methods would not have enough precision to pick up that the zero is off the critical line by such a small amount. How can numerical methods guarantee validity if it is only using some finite number of digits precision?

Answer 1

The zeros of the Riemann zeta function are symmetric about the critical line. So if $1/2+\epsilon+iy_0$ is a zero, so is $1/2-\epsilon+iy_0$. Methods for finding zeta zeros involve computing numerically the number $N(y)$, the number of zeta zeros $z$ with $0<\Im(z)<y$. If we had an $\zeta(1/2+\epsilon+iy_0)=0$ as above, then $N(y)$ would jump by $2$ when passing through $y=y_0$. But all jumps observed in $N(y)$ so far have been by $1$, indicating the presence of a simple zero on the critical line.

What would be difficult to do is to distinguish a double zero of $\zeta$ on the critical line from two simple zeros near the critical line.