## Applying the Explicit Formula

It’s quite some time since we arrived at Riemann’s main result, the explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where $$J(x)$$ is the prime power counting function introduced even earlier. It’s high time we applied this! First, let’s take a look at $$J(x)$$ when calculating it exactly: You see how this jumps by one unit at prime values ($$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, $$17$$, $$19$$), by half a unit at squares of primes ($$4$$, $$9$$), by a third at cubes ($$8$$), and by a quarter at fourth powers ($$16$$), but is constant otherwise. [Read More]