  In previous work it was shown that if certain series based on sums over
primes of non-principal Dirichlet characters have a conjectured random walk
behavior, then the Euler product formula for its $L$-function is valid to the
right of the critical line $\Re (s) > \tfrac{1}{2}$, and the Riemann Hypothesis
for this class of $L$-functions follows. Building on this work, here we propose
how to extend this line of reasoning to the Riemann zeta function and other
principal Dirichlet $L$-functions. We apply these results to the study of the
argument of the zeta function. In another application, we define and study a
1-point correlation function of the Riemann zeros, which leads to the
construction of a probabilistic model for them. Based on these results we
describe a new algorithm for computing very high Riemann zeros, and we
calculate the googol-th zero, namely $10^{100}$-th zero to over 100 digits, far
beyond what is currently known.
