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I found myself wondering the other day whether the fast-growing functions from natural to naturals that are studied by people like proof theorists are the restriction to the naturals of analytic functions from the complexes to the complexes. It turns out, because of the Pringsheim Interpolation formula, that they all are. Now the location of the zeroes of the Riemann zeta function contains lots of information about primes. Might it be the case that the location of zeroes of some of these analytic continuations of fast-growing functions contains information about countable ordinals? Is there an obvious reason why they can't? I do wonder why no connection between this material and complex analysis has ever been made....

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Great idea! Could you say exactly which class of fast - growing functions on the natural numbers you mean? – Joel David Hamkins Mar 26 '14 at 11:53
I presume this is precisely NOT relevant's_theorem – Jp McCarthy Mar 26 '14 at 13:13
I didn't know Carlson's theorem, so thanks! In answer to Joel, i would say - for example - the Hardy Hierarchy. That sort of thing. My puzzlement is over where there is nothing in the theory of this functions that parallels the way in which complex analysis helps in number theory. – Thomas Forster Mar 26 '14 at 20:07
Thomas, even if not exactly your question, you may want to look at the work of Andreas Weiermann and his collaborators, such as Phase transitions in proof theory, Lev Gordeev and Andreas Weiermann. Discrete mathematics and theoretical computer science, (2010), 343-358. – Andrés E. Caicedo Mar 28 '14 at 3:54
Thank you Andres. That does indeed like more-or-less exactly what i'm after.... Hmmmmm – Thomas Forster Mar 30 '14 at 16:01

These zeros can be literally anywhere, by the result you quote that for any discrete set $\{z_n\}$ and any values $a_n$, there are entire functions with $f(z_n)=a_n$.

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Here is an immediate reply to your question, cut and pasted from the Wikipedia article on the factorial function:

The Pi function is certainly not the only way to extend factorials to a function defined at almost all complex values, and not even the only one that is analytic wherever it is defined. Nonetheless it is usually considered the most natural way to extend the values of the factorials to a complex function.

That is, there is no unique way to do what you want, even for the factorial function. There is not even a universally agreed upon best way, or agreement upon what "best" might mean.

To add content to your question, I think a few examples are called for, and perhaps a sensible restriction on the allowed extensions.

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