Zeroes of Functions of Finite Order

Theorem
Let $f(z)$ be an entire function which satisfies $f(0) \neq 0$, and


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

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 $N(T) \ll \alpha(2T)$.

Corollary
If $f$ has order $1$, then for all $\epsilon > 0$, the sum


 * $\displaystyle \sum_{k \geq 1} \frac 1{|\rho_k|^{1+\epsilon}}$

converges, where $\{ \rho_k \}_{k \geq 1}$ is a non-decreasing enumeration of the zeros of $f$, counted with multiplicity.

Proof of Theorem
Fix $T \ge 1$ 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

and

Moreover, by hypothesis we have that


 * $\displaystyle \frac 1{2\pi} \int_0^{2\pi} \log |f(Te^{i\theta})|\ d\theta \leq \sup_{\theta \in [0,2\pi)}\log |f(Te^{i\theta})| \ll \alpha(T)$

Putting these facts into Jensen's formula we have


 * $\displaystyle N\left(\frac T2 \right) \log 2 + |f(0)| \ll \alpha(T)$

Which implies


 * $N(T) \ll \alpha(2T)$

Proof of Corollary
Let $\epsilon > 0$, $N(0) = 0$, so that


 * $\displaystyle \sum_{k\geq 1} |\rho_k|^{-1-\epsilon} \leq \sum_{T \geq 1} \left[ N(T) - N(T-1) \right] T^{-1-\epsilon}$

We have $N(T) \ll 2T$, so $N(T) - N(T-1)$ is bounded in $T$, say by $C > 0$. Therefore,


 * $\displaystyle \sum_{k\geq 1} |\rho_k|^{-1-\epsilon} \leq C\: \sum_{T \geq 1} \frac 1{T^{1+\epsilon}}$

and the sum on the right converges absolutely for $\epsilon > 0$.