On the qth convergence of exponent of entire functions (Q1077582)

Let f(z) be an entire function. The author defines in the distribution of zeros, qth convergence exponent, as \(\rho (q)=g.l.b\). \(\{\alpha =\alpha >0\) and \(\sum^{\infty}_{n=1}(\log^{[q]} r_ n)^{-\alpha}<\infty \}\) where \(0\leq q<\infty\), \(\log^{[0]} (x)=x,\) \((\log^{[q]} (x)=\log (\log^{[q-1]} x)\quad for\quad 0<\log^{[q-1]} x<\infty.\) Theorem 1. If f(z) has at least one zero, then \[ \limsup_{r\to \infty}\log n(r)/\log^{[q-1]} r=\rho (q), \] where n(r) has the usual meaning.