Nevanlinna characteristics and defective values of the Weierstrass zeta function (Q765551)

The authors study the value distribution of the Weierstrass zeta function \(\zeta\), proving the following growth estimates: \[ \begin{aligned} N(r,\zeta) & = \frac{\pi r^2}{2D} + O(r),\\ m(r,\zeta) & = O(\ln r),\\ T(r,\zeta)& = \frac{\pi r^2}{2D} + O(r)\end{aligned} \] when \(r\to\infty\), where \(D\) denotes the area of the primitive period parallelogram of the Weierstrass elliptic function \(\varrho\) (and \(N\), \(T\), \(m\) denote the usual growth functions in Nevanlinna theory). The proof follows basically from the relation \(\zeta = \sigma'/\sigma\) with the Weierstrass sigma function. Further, the authors prove that the Nevanlinna defect of \(\zeta\) at \(\infty\) equals zero and that \(\zeta\) is a meromorphic function of order 2.