Let \(\Gamma\) be a Fuchsian group, cofinite but not cocompact, and \(\omega\) a holomorphic cusp 1-form on \(M=\Gamma \backslash \mathbb H^2\). The modular symbols \(\langle \gamma,\omega\rangle= \int_\gamma\omega\), \(\gamma\in\Gamma\simeq\pi_1(M)\) appear in number theory (e.g. the conjecture of Goldfeld and Szpiro, see D. Goldfeld [Number theory in progress, Vol. 2, Zakopane 1997, de Gruyter, 849–865 (1999; Zbl 0948.11022)]). To study their asymptotic distribution, D. Goldfeld [Proc. Symp. Pure Math. 66, Part 1, 111–121 (1999; Zbl 0934.11026)] introduced an Eisenstein series \(E_{\omega}(z,s),z\in\mathbb H^2,s\in\mathbb C\), whose residue at \(s=1\) is important. C. O’Sullivan [J. Reine Angew. Math. 518, 163–186 (2000; Zbl 0941.11022)] proved some properties of these Eisenstein series: analytic continuation, identification of poles with eigenvalues of the Laplace-Beltrami operator, functional equation.
The Eisenstein series \(E_\omega\) is not an automorphic form. However, the author observes here that \(E_\omega\) is the derivative with respect to \(\varepsilon\) of the Eisenstein series \(E_\varepsilon\) associated to the (non unitary) character \(\chi_\varepsilon: \gamma \to \exp(-2\text{i}\pi \varepsilon \int_\gamma \omega)\) of \( \Gamma\). The paper concentrates on the property of these Eisenstein series \(E_\varepsilon\) whose meromorphic extension is established along the way introduced by Y. Colin de Verdière [C. R. Acad. Sci., Paris, Sér. I 293, 361–363 (1981; Zbl 0478.30035)] and analytic properties are explained with the help of Y. Colin de Verdière’s pseudo-Laplacians [Ann. Inst. Fourier 33, 87–113 (1983; Zbl 0496.58016)] or methods used by R. Phillips and P. Sarnak [e.g., J. Am. Math. Soc. 5, 1–32 (1992; Zbl 0743.30039)]. The main results of C. O’Sullivan (loc. cit.) about \(E_\omega\) are deduced from the properties of \(E_\varepsilon\). Bounds for the Eisenstein series \(E_\omega\) on vertical lines are also obtained.