Zeroes of Functions of Finite Order/Corollary

Corollary to Zeroes of Functions of Finite Order
Let $\map f z$ be an entire function which satisfies:
 * $\map f 0 \ne 0$
 * $\cmod {\map f z} \ll \map \exp {\map \alpha {\cmod z} }$

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

Let $f$ have order $1$.

Let $\size {\rho_k}_{k \mathop \ge 1}$ be a non-decreasing enumeration of the zeroes of $f$, counted with multiplicity.

Then for all $\epsilon > 0$, the summation:


 * $\ds \sum_{k \mathop \ge 1} \frac 1 {\size {\rho_k}^{1 + \epsilon} }$

converges.

Proof
Let $\epsilon > 0$, $\map N 0 = 0$, so that:


 * $\ds \sum_{k \mathop \ge 1} \size {\rho_k}^{-1 - \epsilon} \le \sum_{T \mathop \ge 1} \paren {\map N T - \map N {T - 1} } T^{-1 - \epsilon}$

We have $\map N T \ll 2 T$, so $\map N T - \map N {T - 1}$ is bounded in $T$, say by $C > 0$.

Therefore:


 * $\ds \sum_{k \mathop \ge 1} \size {\rho_k}^{-1 - \epsilon} \le C \ \sum_{T \mathop \ge 1} \frac 1 {T^{1 + \epsilon} }$

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