Zeroes of Functions of Finite Order

Theorem
Let $f(z)$ be an entire function which satisfies


 * $|f(z)| \ll \exp\left( \alpha(|z|) \right)$

for all $z \in \C$ and some function $\alpha$, where $\ll$ is the order notation.

For $T \geq 1$, let


 * $N(T) = \# \{\rho \in \C : f(\rho) = 0,\ |\rho| < T \}$

where $\#$ denotes the cardinality of a set.

Then $N(T) \ll \alpha(T)$.

Proof
Fix $T \geq 1$ and let $\rho_1,\rho_2,\ldots,\rho_n$ be an enumeration of the zeros of $f$ with modulus less than $T$, counted with multiplicity.

By Jensen's Formula we have


 * $\displaystyle \frac 1{2\pi} \int_0^{2\pi} \log |f(Te^{i\theta})|\ d\theta = \log|f(0)| + \sum_{k=1}^n (\log T - \log |\rho_k|)$

Let $\rho_0 = 1$, $\rho_{n+1} = T$, $r_k = |\rho_k|$. Then