Let $M$ be a finite volume hyperbolic Riemann surface with arbitrary signature, and let $\chi$ be an arbitrary $m$-dimensional multiplier system of weight $k$. Let $R(s,\chi)$ be the associated Ruelle zeta function, and $\varphi(s,\chi)$ the determinant of the scattering matrix. We prove the functional equation that $R(s,\chi)\varphi(s,\chi) = R(-s,\chi)\varphi(s,\chi)H(s,\chi)$ where $H(s,\chi)$ is a meromorphic function of order one explicitly determined using the topological data of $M$ and of $\chi$, and the trigonometric function $\sin(s)$. From this, we determine the order of the divisor of $R(s,\chi)$ at $s=0$ and compute the lead coefficient in its Laurent expansion at $s=0$. When combined with results by Kitano and by Yamaguchi, we prove further instances of the Fried conjecture, which states that the R-torsion of the above data is simply expressed in terms of $R(0,\chi)$.

Abstract In [20], Rohrlich proved a modular analog of Jensen’s formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$ . In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$ , thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$ , where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two.

Let X be a smooth, compact, projective Kähler variety and D be a divisor of a holomorphic form F , and assume that D is smooth up to codimension two. Let ω be a Kähler form on X and KX the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on X. Using various integral transforms of KX , we will construct a meromorphic function in a complex variable s whose special value at s = 0 is the log-norm of F with respect to μ. In the case when X is the quotient of a symmetric space, then the function we construct is a generalization of the so-called elliptic Eisenstein series which has been defined and studied for finite volume Riemann surfaces.

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved an analogue of Kronecker’s limit formula associated to any divisorD which is smooth in codimension one on any smooth Kähler manifold X . In the present article, we apply the aforementioned Kronecker limit formula in the case when X is complex projective space CP for n ≥ 2 and D is a hyperplane, meaning the divisor of a linear form PD(z) for z = (Zj) ∈ CP. Our main result is an explicit evaluation of the Mahler measure of PD as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L-norm of the vector of coefficients of PD.