# Month: January 2020

• ## Dirichlet’s Prime Number Theorem – Part 2

The next step in this series of blog posts, which started with the first part, ist to show why the Riemann Zeta function does not diverge at $s=2$ and why its value at this point is $\frac{\pi^2}{6}$.

I recently found 2 ways to show that the infinite sum of all reciprocals of perfect squares is $\frac{\pi^2}{6}$:

• The first way is via constructing a representation of the $\sin$ function from its roots and then comparing it to its well-known Taylor series. I first found this proof in the mathologer video https://www.youtube.com/watch?v=yPl64xi_ZZA. [Sullivan2013] elaborates in more detail on that solution.
• The second way is via the total amount of apparent brightness received from an array of light sources arranged equidistantly around an infinitely large circle. ( https://www.youtube.com/watch?v=d-o3eB9sfls)

I will only discuss the second proof in this post whose proof is based on the paper at [Wästlund2010]. The proof presented here is identical to that one, I have even copied the figures from that paper. However, I have tried to put in some more details and make it easier to read for someone who encounters this for the first time.

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