Atsuji proposed some integrals along Brownian paths to study the Nevanlinna characteristic function when is meromorphic in the unit disk . We show that his criterion does not apply to the basic case when is a modular elliptic function. The divergence of similar integrals computed along the geodesic flow is also proved.
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Jean-Claude Gruet, Nevanlinna theory, fuchsian functions and brownian motion windings. Rev. Mat. Iberoam. 18 (2002), no. 2, pp. 301–324DOI 10.4171/RMI/320