Given an Arithmetic function, (or even better who's values are integers), how can I tell if it has an Analytic continuation to the whole plane, or maybe half plane?
I guess it might be too general a question.
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$\begingroup$ this is not what you are asking for but you may look into en.wikipedia.org/wiki/Nevanlinna%E2%80%93Pick_interpolation $\endgroup$– KoushikDec 9, 2013 at 6:38
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There are many such analytic continuations (though almost surely none of them interesting). You can always find an entire function taking whatever sequence of values you want. More precisely, given a sequence $\{z_1,z_2,\dots,\}$ of distinct complex numbers such that $|z_j|\to\infty$, and a second sequence $\{w_1,w_2,\dots\}$ of complex numbers, there exist uncountably many entire functions $f$ such that $f(z_j)=w_j$ for each $j$. So for example, there exist uncountably many entire functions $f$ such that $f(n)=\phi(n)$ for each positive integer $n$.
answered Dec 9, 2013 at 6:30
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2$\begingroup$ I see, it's existence is an application of one theorem in Complex analysis. It's sad, since I hope to check the property of the function by considering it's Analytic continuation. One further question, how can I give a Analytic continuation explicitly? Or some interesting Analytic continuation. $\endgroup$ Dec 9, 2013 at 6:45
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