Does there exist a non-constant entire function $f : \mathbb{C}\to\mathbb{C}$ such that $f(n+\dfrac{1}{n})=0$ for all $n\in \Bbb N$?
Let $f$ be a non-constant entire function such that $f(n+\dfrac{1}{n})=0\forall n\in \Bbb N$.
Then $f(2)=0;f(3+\frac{1}{3})=0$ and so on.But the problem is the set of zeros of $f$ does not have a limit point.
How can I conclude whether such a function exists or not?Please help
Best Answer
There exists such a function. An infinite product such as
$$f(z) = \prod_{n =1}^\infty \left(1-\frac{z^2}{(n+1/n)^2}\right)$$
determines a nonconstant entire function of $z$ with zeroes at $z= \pm (n+1/n)$.