Zeros of Functions of Finite Order

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Theorem

Let $f \left({z}\right)$ be an entire function which satisfies:

$f \left({0}\right) \ne 0$
$\left\vert{f \left({z}\right)}\right\vert \ll \exp\left({\alpha \left({\left\vert{z}\right\vert}\right)}\right)$

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


For $T \ge 1$, let:

$N \left({T}\right) = \# \left\{{\rho \in \C: f \left({\rho}\right) = 0, \ \left\vert{\rho}\right\vert < T}\right\}$

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


Then:

$N \left({T}\right) \ll \alpha \left({2 T}\right)$


Corollary

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

$\displaystyle \sum_{k \mathop \ge 1} \frac 1 {\left\vert{\rho_k}\right\vert^{1 + \epsilon} }$

converges, where $\left\langle{\rho_k}\right\rangle_{k \mathop \ge 1}$ is a non-decreasing enumeration of the zeros of $f$, counted with multiplicity.


Proof

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:

$\displaystyle \frac 1 {2 \pi} \int_0^{2 \pi} \log \left\vert{f \left({T e^{i \theta} }\right)}\right\vert \ \mathrm d \theta = \log \left\vert{f \left({0}\right)}\right\vert + \sum_{k \mathop = 1}^n \left({\log T - \log \left\vert{\rho_k}\right\vert}\right)$


Let $\rho_0 = 1$, $\rho_{n+1} = T$, $r_k = \left\vert{\rho_k}\right\vert$.

Then:

\(\displaystyle \int_0^T N \left({t}\right) \frac{\mathrm d t} t\) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n \int_{r_k}^{r_{k+1} } N \left({t}\right) \frac {\mathrm d t} t\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^n k \log \left( \frac{r_{k+1} } {r_k} \right)\) as by the definition of $N$, it is constant value $k$ on each interval $\left({\left\vert{\rho_k}\right\vert \,.\,.\, \left\vert{\rho_{k+1} }\right\vert}\right)$
\(\displaystyle \) \(=\) \(\displaystyle \log \left({\frac {T^n} {r_1 \cdots r_n} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \left({\log T - \log r_k}\right)\)

and

\(\displaystyle \int_0^T N \left({t}\right) \frac {\mathrm d t} t\) \(=\) \(\displaystyle \int_0^2 N \left({\frac{T \theta} 2}\right) \frac {\mathrm d \theta} {\theta}\)
\(\displaystyle \) \(\ge\) \(\displaystyle N \left({\frac T 2}\right) \int_1^2 \frac {\mathrm d \theta} {\theta}\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle N \left({\frac T 2}\right) \log 2\) Definition of Logarithm


Moreover, by hypothesis we have that:

$\displaystyle \frac 1 {2 \pi} \int_0^{2 \pi} \log \left\vert{f \left({T e^{i \theta} }\right)}\right\vert \ \mathrm d \theta \le \sup_{\theta \mathop \in \left[{0 \,.\,.\, 2 \pi}\right)} \log \left\vert{f \left({T e^{i \theta} }\right)}\right\vert \ll \alpha \left({T}\right)$


Putting these facts into Jensen's formula we have:

$N \left({\dfrac T 2}\right) \log 2 + \left\vert{f \left({0}\right)}\right\vert \ll \alpha \left({T}\right)$

which implies:

$N \left({T}\right) \ll \alpha \left({2 T}\right)$

$\blacksquare$


Proof of Corollary

Let $\epsilon > 0$, $N \left({0}\right) = 0$, so that

$\displaystyle \sum_{k \mathop \ge 1} \left\vert{\rho_k}\right\vert^{-1-\epsilon} \le \sum_{T \mathop \ge 1} \left({N \left({T}\right) - N \left({T - 1}\right)}\right) T^{-1 - \epsilon}$

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

Therefore:

$\displaystyle \sum_{k \mathop \ge 1} \left\vert{\rho_k}\right\vert^{-1-\epsilon} \le C \ \sum_{T \mathop \ge 1} \frac 1 {T^{1 + \epsilon} }$

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

$\blacksquare$