My question is: What are the hypothetical zeros of the Dirichlet eta function (the alternating zeta function) in the critical strip. This notion is far from my understanding.
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Well, since the Dirichlet eta function is the analytic continuation of the zeta function in the critical strip, its zeros there are precisely the zeros of the zeta function there and, according to the R.H., they all have real part equal to $\,0.5\,$ ...