It is a relatively well-known fact that Random Walks are a pretty good approximation of the locus of partial sums for the zeta function.
For instance let $\zeta_k(s)$ denote the $k^\text{th}$ partial sum of the Riemann zeta function:
$$ \zeta_k(s) := \sum_{n=1}^k n^{-s} $$
Then we can consider the image of the partial sums up to $k$ by $Z_k=\{\zeta_m(s):1\leq m\leq k \}$.
Now I don't know very much about random walks; my knowledge of them can be summarized by: "A random walk is a random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers" as per the Wikipedia definition. Never the less I know what their graphs can look like.
This random walk is a representation of Brownian Motion for example.
I was playing around with the locus of partial sums and the thought occurred to me that these to objects may in fact be related since they look very similar. Computing $\zeta_k(\frac12 + i1.5\times 10^9)$ and computing the locus for $Z_{12500}$ gives the following graph:
There is clearly some resemblance here between these two objects. As it turns out I was not the first to notice this and as it turns out it is fairly well known that certain random walks make excellent approximations for the locus of zeta function partial sums with a sufficiently high imaginary value. My question here is why? Random Walks are by their very nature random, but the zeta function is absolutely deterministic? How can we model random walks with something that is absolutely not random? Can anyone offer some kind of intuition for this? Or any suggested readings?