Testing Zeros Of The Riemann Hypothesis I was on Mathworld some time ago when I read this from http://mathworld.wolfram.com/RiemannHypothesis.html:
The Riemann hypothesis was computationally tested and found to be true for the first 200000001 zeros by Brent et al. (1982), covering zeros sigma+it in the region 0 < t < 81702130.19.
My question is: How can you be sure that you haven't missed any zeros? It seems to me that it is impossible because for any fixed t one would have to check all real sigma values between 0 and 1. And even if there was some way to do that one would still need to test all real values of t between 0 and 81702130.19. Do they have a list of "candidate zeros" that they would just try out?
Thanks in advance.
 A: Let $N(T)$ be the number of non-trivial zeros up to height $T$ : $N(T) = \#\{ \rho \ \mid \ 0 < Im(\rho) < T \ \}$ and $N_0(T)$ those lying on the critical line. The Riemann hypothesis is that $N(T) = N_0(T)$ for every $T$.


*

*you need to understand the functional equation $\xi(s) = \xi(1-s)$ where $\xi(s) = A(s) \zeta(s)$ and $A(s) = \frac{1}{2}s (s-1) \pi^{-s/2} \Gamma(s/2) $. Together with $\xi(s) = \overline{\xi(\overline{s})}$ it shows that $\xi(1/2+it)$ is real. Hence it has one zero at every sign change.

*and the argument principle showing that $$2 N(T) = \frac{1}{2i\pi} \oint_{\begin{array}{l}2- i T\to 2+ i T \to\\  -1+iT \to -1-iT \to 2-iT\end{array}} \frac{\xi'(s)}{\xi(s)}ds = \frac{2}{\pi}\text{arg } A(1/2+iT) + \frac{2}{\pi} \text{arg } \zeta(1/2+iT)$$
where $\text{arg } f(s) = \text{Im}(\log f(s))$ is defined by starting with $\text{arg } f(2) = 0$, and following $\log f(s)$  analytically on $2+it, t \in [0,T]$, and then on $\sigma+iT,\sigma \in [2,1/2]$ (assuming $f(s)$ has no zero on $Re(s) > 1$ and $Im(s) = T$ and that $f(2) > 0$)
Everything is explained for example in Titchmarsh's book "the theory of Riemann zeta function", and how to estimate all these in practice for the first few zeros, using the function $Z(t)$. At the end, you can bound $N(T)$ and $N_0(T)$ within an accuracy $< 1/2$, and show they are equal.
