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

How NOT to Earn a Million Dollars

I recently spent some timeĀ 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:

Continue reading How NOT to Earn a Million Dollars