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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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Perhaps this recent paper will of interest to you regarding All the zeros of the Dirichlet eta function in the critical strip are on the critical line. Regards –  Amzoti Jan 3 '13 at 16:06
    
@ Amzoti: I will read it. Thank you very much. –  ZE1 Jan 3 '13 at 16:10
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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 $\,0.5\,$ ...

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@ DonAntonio: I found this now: and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of rectangles symmetrical around the x-axis and the critical line and whose multiplicity is unknown. Plese see en.wikipedia.org/wiki/Dirichlet_eta_function –  ZE1 Jan 3 '13 at 16:08
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