Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ?
Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ?
Im not looking for the positions of the zero's nor closed form for them.
I just want the amount of zero's for a general real entire function.
Im aware of the Argument Principle, Rouché 's theorem and many other theorems. But it seems to help little for the general case.
I also wonder; in number theory its often the position of the zero's that matters. What is a typical example of when the amount matters more (in number theory) ?