Tag Archives: Approximation

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:

J(x) Continue reading Applying the Explicit Formula