It's quite some time since we arrived at Riemann's main result, the explicit formula
where is the prime power counting function introduced even earlier. It's high time we applied this!
First, let's take a look at when calculating it exactly:
I recently spent some time on the formidable website Numberphile which explains mathematical ideas, some important, some recreational, in short and accessible videos. Definitely worth checking out. One of the videos that is most relevant to us explains the Riemann Hypothesis: