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In this proof, the election of the riemann function in order to perform the analytic continuation seems just like one of the infinite functions we can choose.

So the questions would be:

1) if we choose other functions in order to perform the analytic continuation, can we obtain other results? (For example 174). Can you give examples?

2a) If that's the case, why in several branch of physics, the "-1/12 result" is used. What characteristic does -1/12 has that physicists tend to use it more than the rest of the numbers?

2b) if it is not the case, where is the demostration that all "analytic continuations proof" of this sum give -1/12?

Thanks in advance for any feedback.