Constructing an entire function

Question

This is a question from my complex analysis final exam: Does there exists an entire function $f$ such that $f(\log k)=1/\log k$ for all $k\geq 2$, integer. My answer is a no. What do you guys think?

Answer 1

It matters here that the set is discrete. In general you cannot specify the values of an entire function on a continuum, nor on, for example, $\{ 1/n : n \in \mathbb{N} \}$ (a set with a finite limit point). But when $\{ z_n \}_{n=1}^\infty$ is discrete, for any sequence $\{ y_n \}_{n=1}^\infty$ there is an entire function $f$ with $f(z_n)=y_n$. When $y_n \equiv 0$ for instance, the Weierstrass factorization theorem will do the job for you.