convergence exponent

**tenderline** · January 11th 2013, 02:45 AM

Let $\{z_j\}$ be the sequence of zeros on an entire function $f$ . We define the convergence exponent of $\{z_j\}$ as
$b=\inf\left\{\lambda>0\ \text{s.t.}\ \sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\lambda}}\ \textrm{converges}\right\}$

Let $n(r)$ be the number of $z_j$ 's with $|z_j|\leq r$ . Then the following identity holds:
$b=\limsup_{r\rightarrow +\infty}\frac{\log{\ n(r)}}{\log{r}}$

Do you think i should use Jensen formula to prove this?

**choovuck** · January 20th 2013, 01:49 PM

don't know if Jensen would help or not, but the proof that I have seen uses formula

$\sum \frac{1}{|z_j|^\lambda} = \int_0^\infty \frac{dn(t)}{t^\lambda}$

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