The simplest explanation of what the Riemann Hypothesis means

The logarithmic integral function $\mathrm{li}(x)$ is defined as

\[ \mathrm{li}(x) = \int_0^{\infty} \frac{\mathrm{d}t}{\log(t)} \]

where $\log$ is is the natural logarithm, and the integral is understood as the Cauchy principal value at $t=1$. The function $\mathrm{li}(x)$ is for $x>1$ a nice and smooth function which grows roughly like $\frac{x}{\log(x)}$ as $x$ goes to infinity. See Wikipedia for a plot and more details.

The prime-counting function $\pi(x)$ is defined as the number of primes up to and including $x$, so for example, $\pi(10) = 4$ and $\pi(1000) = 168$.

Often the Riemann Hypothesis is explained in terms of a statement using big-O notation, so that there is an implied constant left unspecified. But a more explicit and hence more elementary formulation of the Riemann hypothesis is that $\pi(x)$ can be approximated by $\mathrm{li}(x)$, in precise sense that the inquality

\[ \vert \pi(x) - \mathrm{li}(x) \vert < \frac{\sqrt{x} \log(x) }{8 \pi} \]

holds true for all $x>2657$.

Exercise: What goes wrong for $x$-values smaller than 2657? Does the inequality fail for all $x \leq 2657$ or only in specific ranges? Does it fail because there are "too many primes", or "too few"?

Exercise: What do you think happens with the ratio between the left hand side and the right hand side as $x$ goes to infinity?

Below is some code to get started.