Speeding through the primes until they look uniform Imagine you are traveling out the $x$-axis such that your velocity at $x$ is $v(x)$.
If $v(x)=1$, then you pass the primes at increasingly longer intervals, on average (of course there are close primes, e.g., twin primes).
Is there some function $v(x)$ so that the event of passing a prime becomes uniformly random,
in that the expected time between passing primes becomes a constant?
For example, $v(x) = x / \log^2 x$ doesn't quite work, in that the delay between
passing primes still increases with $x$.

A tautological solutions would be variations of the step function
$$
v(x) = 1/d(p_x)
$$
where for each $x$, $p_x$ is the biggest prime less than or equal to $x$ and $d(p)$ is the distance between the prime $p$ and the next prime. Of course, we can turn this into a smooth function by multiplying by suitable bump functions (and adjusting the levels of the steps slightly, so as to insure that the time it takes to travel between two primes stays the same).
Unfortunately, a non-tautological solution, even conjectural, that insures that the time intervals between two primes on average converge to a constant value is completely out of reach. If we had such a closed form formula for the velocity, then in particular it would yield an approximate formula for the prime counting function $\pi(x)$ whose error term tends to 0. This is not even known under the GRH and might not even exist.
