Geodesic
Nielsen zeta-function
Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part V, Tome 143 (1985), pp. 156-161

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In this paper we introduce a new zeta-function in the theory of dynamical systems. We find a sharp bound for the radius of convergence of the Nielsen zeta-function in terms of the topological entropy of the map. It follows from this that the Nielsen zeta-function has a positive radius of convergence. We prove that for an orientation-preserving homeomorphism of a compact surface the Nielsen zeta-function is either a rational function or the radical of a rational function. We calculate the Nielsen zeta-function for maps of circles, spheres, tori, protective spaces, for expanding maps of an orientable smooth compact manifold, for a homotopy periodic map of a connected compact polyhedron having no locally separating point. 
@article{ZNSL_1985_143_a8,
     author = {V. B. Pilyugina and A. L. Fel'shtyn},
     title = {Nielsen zeta-function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {156--161},
     publisher = {mathdoc},
     volume = {143},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_143_a8/}
}
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V. B. Pilyugina; A. L. Fel'shtyn. Nielsen zeta-function. Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part V, Tome 143 (1985), pp. 156-161. http://geodesic.mathdoc.fr/item/ZNSL_1985_143_a8/