A pole for F(s) means zeta(s) = 0 but zeta(2s) not=0 for some complex s. So yes, the zeros on the critical line appear as poles in F(s). Looking outside the cri

A pole for F(s) means zeta(s) = 0 but zeta(2s) not=0 for some complex s. So yes, the zeros on the critical line appear as poles in F(s). Looking outside the critical line, at negative even integers, you will get 0/0 and using L-Hospitals rule, they may end up cancelling out, meaning F(s) remains analytic at these locations. But my point is how do you know this cancellation isn't occurring at other locations, outside the critical line? I am not saying there are other poles outside the critical line, I am saying the author hasn't ruled out 0/0 happening in the critical strip. This is really just high school calculus we are talking about, 0/0 is an indeterminate form and more work must be done to see what type of singularity it is. The author saying F(s) is analytic outside the critical line but within the critical strip means there could still be removable singularities, i.e. 0/0, that end up cancelling out. These are like holes in our graphs back in Cal I. Riemann's Hypothesis is 157 years old now, it is not going to be proven by a 20 page paper.