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Did you use Euler's definition that the zeta of 2n is equal to Bernoulli of 2n times 2pi to the 2n over 2 times 2n factorial times minus 1 to the n+1th power?
That is a big statement. Are you sure 4 is correct?
To others, infinity is a concept not a number but to me there is even and odd infinity. To them 1+infinity=infinity but to me 1+infinity>infinity. They got all those values like 1/12 or etc because they play with the infinity as they like. If you have a function of 1-1+1-1+1..infinity, if you use their concept you would get sometimes S=1/2 yet you know when it is even, S=0 and when it is odd, S=1 and this function alternates 0 & 1 to the infinity. I think people need to respect the infinity, otherwise we would be hay-wired. I do sometimes play with the infinity and I can proof that Zeta functionis not always true and converge to value 4.
I read the article. So the definition changes for that domain with gamma functions and what not.
only when the sum converges which it clearly does not for s=0. So the above definition does not hold. Look here for how an analytic continuation is used:
Alright, ζ(0) = 1 + 1 + 1 + ... = -1/2. Miracles do happen!