convergence exponent

• Jan 11th 2013, 03:45 AM
tenderline
convergence exponent
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?
• Jan 20th 2013, 02:49 PM
choovuck
Re: convergence exponent
don't know if Jensen would help or not, but the proof that I have seen uses formula

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