Atsuji proposed some integrals along Brownian paths to study the Nevanlinna characteristic function $T(f,r)$ when $f$ is meromorphic in the unit disk $D$. We show that his criterion does not apply to the basic case when $f$ is a modular elliptic function. The divergence of similar integrals computed along the geodesic flow is also proved.
"Nevanlinna theory, fuchsian functions and brownian motion windings." Rev. Mat. Iberoamericana 18 (2) 301 - 324, June, 2002.