On primeness of the Selberg zeta-function (Q1996198)

The paper under review studies the Selberg zeta-function \(Z(s)\) associated with a compact Riemann surface of genus \(g\). The main results states that \(Z(s)\) is pseudo-prime and right-prime. More precisely, for every decomposition \(Z(s)=f(h(s))\) with \(f\) meromorphic and \(h\) entire (or \(h\) meromorhic when \(f\) is rational), the following hold: (1) \(f\) is rational or \(h\) is a polynomial; (2) \(h\) is linear whenever \(f\) is transcendental (noting a typo in the Definition, p. 452). Moreover, if \(f\) is rational and \(h\) is meromorphic, then \(f\) is a polynomial of degree \(k\) where \(k\) divides \(2g-2\), and \(h\) is entire.