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Let $\Gamma$ be a discrete subgroup of $SL(2, \doubz$) with fundamental domain of finite invariant measure ${dydx\over y\sp2}$. $\Gamma$ acts discontinuously on the complex upper half-plane $\Pi$. This action can be extended to $\IR \cup \{\infty\}$. To each cusp $k\sb{i}$ of $\Gamma$ L. Goldstein in (1) attaches an eta function $\eta\sbsp{\Gamma}{k\sb{i}}$. Then $\eta\sbsp{SL(2,\doubz)}{\infty}$ coincides with the classical Dedekind eta function. It is well known that the Eisenstein series$$E\sbsp{\Gamma,s}{k\sb{i}}(z)=\sum\sb\sigma y(\sigma\sbsp{i}{-1}\sigma z)\sp{s}$$(where the sum is over a complete set of representatives $\sigma$ of the cosets $\Gamma\sb{k\sb{i}}\\\Gamma, \Gamma\sb{k\sb{i}}$ is the stabilizer of $k\sb{i}$ in $\Gamma, Re(s) >$ 1 and $z \in\ \Pi)$ converges absolutely and uniformly for s in any compact subset of $Re(s) >$ 1. This series can be analytically continued to a meromorphic function in the entire s-plane. The continllation has all its poles in the interval (0,1) and s = 1 is always a pole. $E\sbsp{\Gamma,s}{k\sb{i}}$ is an automorphic function for $\Gamma$ and an eigenfunction of the Laplace-Beltrami operator. Therefore it can be expanded in a Fourier series at the cusp $k\sb{i}$. For the Eisenstein series for the modular group there exist explicit formulae for the coefficients of this expansion. The generalized function $\eta\sbsp{\Gamma}{k\sb{i}}$ satisfies a transformation formula similar to the one satisfied by the Dedekind eta function $\eta\sbsp{SL(2,\doubz)}{\infty}$ for the modular group. This formula involves a generalization $S\sbsp{\Gamma}{k\sb{i}}(\gamma)$, where $\gamma \in\ \Gamma$, of the Dedeklnd sum. We consider the structure of $\eta\sbsp{\Gamma\sb0(N)}{\infty}$ and $S\sbsp{\Gamma\sb0(N)}{\infty}$ for the Hecke subgroup $\Gamma\sb0(N)$. A key point for the purposes of studying this eta function is to obtain an explicit Fourier expansion for the Eisenstein series for $\Gamma\sb0(N)$ at the cusp $\infty$. We derive a Kronecker limit formula for this case and explicitly describe $\eta\sbsp{\Gamma\sb0(N)}{\infty}$ and $S\sbsp{\Gamma\sb0(N)}{\infty}$. Knowing the generalized Dedekind sum allowed the study of some of its arithmetic properties. In particular, we verify a conjecture of L. Goldstein on the rationality of $S\sbsp{\Gamma\sb0(N)}{\infty}$. In the last chapter we turn to a general cusp $k\sb{i}$ of $\Gamma$ and express the eta function $\eta\sbsp{\Gamma}{k\sb{i}}$ in terms of the eta function at the cusp at infinity. Our results are motivated by, but independent, of L. Goldstein's considerations in (?). |
