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 and why its value at this point is .
I recently found 2 ways to show that the infinite sum of all reciprocals of perfect squares is :
- The first way is via constructing a representation of the 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.(more…)