Take a peek at this very nice paper: http://www.math.titech.ac.jp/~inoue/...lder/zeta2.pdf
You have there no less than 14 different proofs of the result , from rather basic ones to those that require more knowledge....enjoy!
I believe the OP mostly posts challenge problems but places them in the regular forums. This thread has been answered many times over and I am going to PM the OP about using the Challenge Problems forum to save everyone time.