Analytic extension for a a function defined in $\mathbb{N}$

Question

I would like to know if it is possible to extend analytically any function of the type $f:\mathbb{N} \to \mathbb{C}$ to all complex plane. If it isn't possible, what should I assume to do so? If

Just an example: the function number of divisors of $n$.

EDIT: Is it unique?

Answer 1

The answer is yes. First, put $$p_n(z)=\frac{(z-1)(z-2)\cdots(z-n+1)}{(n-1)!}$$ and note that $p_n(k)=0$ for $k=1,2,\ldots,n-1$ while $p_n(n)=1$. Now let $$ f(z)=\sum_{n=1}^\infty b_ne^{a_n(z-n)}p_n(z),\qquad b_n=f(n)-\sum_{k=1}^{n-1}b_ke^{a_k(n-k)}p_k(n). $$ Just make sure that $a_n>0$ is big enough so that, say, $$|f(n)e^{a_n(z-n)}p_n(z)|<2^{-n}$$ for all $z$ with $|z|\le n-1$. Then the series converges uniformly on every bounded set.