(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

In complex analysis analytic continuation is a method that extends a function—when possible—to more values. This extension is one reason we believe that the RH is hard.

The problem that I have with unrestrictedly drawing conclusions about the behaviour of the primes from the behaviour of $\zeta(\sigma + it)$ using analytic continuation is that relationship of $\zeta(\sigma + it)$ to the primes is well-defined only for $\sigma >1$.

In other words, if some function is defined only for points along the $x$-axis for $x>0$, then how can we deduce any behaviour of the function for $x \leq 0$; i.e., in areas where the function does not exist?

Consider, for a moment, the implications of such reasoning for a real world model of such behaviour as exemplified, for instance, by a light signal $S$ which is emitted from the origin along the $x$-axis, so its distance from the origin at time $t>0$ is $x=ct$, where $c$ is the speed of light.

Obviously, the function $x=ct$ is well-defined for values of $t \leq 0$.

However, any such value cannot describe any property of the signal $S$, since $S$ does not exist for $t \leq 0$.

In case this seems far-fetched, Sections 4 to 6 here analyse a Zeno-type example, where even the theoretical limiting behaviour of an elastic string, under a specified iterative transformation, is not given by the putative Cauchy limit (which can only be described as a misleading mathematical myth) of the function that seeks to describe the iteration.

The significance of this is that Hadamard and de la Vallée Poussin’s proof of the Prime Number Theorem draws conclusions about the behaviour of the prime counting function $\pi(n)$, as $n \rightarrow \infty$, by the limiting behaviour of $\zeta(\sigma + it)$ along $\sigma =1$, notwithstanding that such a limit too may be mythical!
Author’s working archives & abstracts of investigations