KU Math 800: Lecture 2/18/2026

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[theorem=%counter% (Theorem 2.1)] If $f$ is entire with order of growth $p$, then

  1. The number of zeros $\pi(r)$ of $f$ in $|z| < r$ satisfies $\pi(r) \le Cr^p$ for a constant $C$ and $r » 0$.
  2. If $z_1, \ldots, z_K, \ldots$ are the zeros of $f$ with $z_K \neq 0$, then $\sum_{k=1}^\infty \frac{1}{|z_k|^s} < \infty$ if $s > \rho$. [/theorem]

[example=%counter%] Recall that

\[\sin(\pi z) = \frac{e^{i\pi z} - e^{-i\pi z}}{2i}.\]

So then

\[|\sin(\pi z)| \le e^{\pi |z|}\]

then $f(z) = \sin(\pi z)$ with zeros at $z\in \ZZ$, satisfies:

  1. $\pi(r) \le r$,
  2. $\sum_{k=1}^\infty \frac{1}{|z_k|^s} < \infty$ if $s > p$. [/example]

[theorem=%counter% (Jensen’s Formula)]

\[\int_{0}^R \pi (x) \frac{\dd{x}}{x} = \frac{1}{2\pi} \int_{0}^{2\pi} \log |f(Re^{i\theta})| \dd{\theta} - \log|f(0)|\]

assuming $f(0) \neq 0$. [/theorem]