One last point. The RH being true implies F(s) is analytic outside the critical line but within the critical strip. If we take what's called the contrapositive

One last point. The RH being true implies F(s) is analytic outside the critical line but within the critical strip. If we take what's called the contrapositive of this statement, then if we could somehow show F(s) is not analytic in this region, then this would disprove the RH. So what the paper claims to be true is a necessary but not sufficient condition for the RH to be true. Since most think the RH does hold, the paper's claim most likely is true and may even have already been proven elsewhere. Clearly the key is understanding what the word analytic means in complex function theory. It's basically like saying a real-valued function in Calculus I is differentiable, but now we are talking about what this means for a complex-valued function.