My question is about the definition of the critical strip: Some authors (http://mathworld.wolfram.com/CriticalStrip.html) consider it as the set of complex numbers $z=x+iy$ with $0<x<1$ and others consider it with the condition $0≤x≤1$ (http://planetmath.org/criticalstrip). Some authors calimed that the Riemann hypothesis is false for the eta function $∑_{n=1}^{∞}(-1)ⁿ⁺¹/n^{z}$ since it have zeros of the form $1+iv$. Some others claimed that the zeta and the eta functions has the same nontrivial zeros on the critical strip and hence the RH is true for the eta function.
How one can understant all these contradictions.
Serre has work exhibiting zero-free regions in the critical strip. These are open regions containing the axes $x=0$ and $x=1$ though they get narrower and narrower for large $y$. At any rate this shows that there are no zeros on the two axes and therefore it is immaterial whether the critical strip is defined by $<$ or $\leq$.