In his popular science book *The Music of the Primes*, Marcus du Sautoy tries to link the truth of the Riemann Hypothesis to the "randomness" of the primes. To do this, he invokes the idea of a "fair coin". This, he claims that probability theory tells us, must satisfy an asymptotic relation whereby the cumulative difference between the number of heads and the number of tails should be $o(x^{1/2})$. He then proposes the difficult-to-visualise idea of a "prime number die" with $\ln n$ sides, so that the probability of each $n$ being prime is $1/\ln n$. And he states that this die will be "fair" if and only if the RH is true. His attempted explanation is necessarily vague and impressionistic, relying on the fact the RH is equivalent to $li(x) - \pi(x)$ being $o(x^{1/2+\epsilon})$ for any positive epsilon (similarly for $\psi(x) - x$, or the Mertens function), and then trying to explain $o(x^{1/2})$ in terms people are familiar with (an unbiased coin toss).

I was wondering if something like this could be made more precise. Suppose we define the increasing sequence $x_k$ for $k =0,1,2,3,...$ where $x_0 = 2$ and $Li(x_{k}) - Li(x_{k-1}) = 0.5$ (that is, $\int_{x_{k-1}}^{x_k} dx/ln x = 0.5$). The idea is that each interval $(x_{k-1},x_k]$ has a 0.5 probability of containing a prime number (taking the density of primes to be $1/ln x$ as usual).

So we then have a "random bit generator": $b_n = 0$ if there's no prime in $(x_{n-1},x_n]$ and $b_n = 1$ if there's at least one prime in the interval.

So would this sequence of bits pass the test for "unbiasedness" which du Sautoy refers to?

We could consider $2H(\pi(x_n)-\pi(x_{n-1})) - 1$ where $H(x)$ is the variation of the Heaviside function which is 1 for positive $x$ and 0 for nonpositive $x$. This will produce the value +1 if there are primes in the interval, and -1 if there are none. So we sum these values and ask: is this $o(x^{1/2})$?

My guess would be that the RH will be equivalent to this function being $o(x^{1/2+\epsilon})$ for any positive $\epsilon$. Any thoughts on this?