One generating function in the rational homotopic type (Q1079224)

Let X be a simply connected CW-complex and \(P_{\Omega X}\) be the Poincaré series of the loop space of X. Denote by \(\gamma_ n\) the rank of \(\pi_{n+1}(X)\) and set \[ (1)\quad T_ X(z)=\sum^{\infty}_{n=1}(-1)^ n \gamma_ n/n^ z. \] The Dirichlet series (1) generally speaking, does not converge and should be understood as a formal series. In the present paper the author shows that if \(P_{\Omega X}\) is a rational function, then the series (1) can be summed up in some reasonable way and its sum can be expressed in terms of the zeta function of Riemann and Hurwitz and the Euler gamma function.