Hi! I just found a limit (maybe someone has found it before) experimentally that
But I can't prove it. Can anyone show a proof to me? Thanks!
I think I have it. Since we are talking about the complex plane we better have a definition of the zeta function in a neighborhood around s = 1. So from mathworld, a series representation is:
So now the above becomes:
From wikipedia we have that
and for z = 2 we have
Hence now we have
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I factored out the sum from the limit since at its limiting value it is just the logarithm of 2 so it does not affect the limit. By L'Hopital's rule the limit is:
Multiplying things out you get the desired expression.
I like your proof. Here is a shorter version of the proof of the limit. Consider the Laurent series of zeta around s =1 :
.
.
Taking the limit as s goes to 1 is now a piece of cake. Although if you didn't have the Laurent series, it'd be more difficult to show.
However, if we are aware of the fact that s-1 is a simple pole, we will immediately get a Laurent series like the above, valid is some neighborhood around s=1 where we don't really care about the coefficients. Then we have the same situation as above.
EDIT: I just realized that interchanging the limit and the sum will require justification. That is we need the series to be uniformly convergent.