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.

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 M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $$\chi $$ χ denote a finite dimensional unitary representation of the fundamental group of M . Let $$\Delta $$ Δ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with $$\chi $$ χ . From the spectral theory of $$\Delta $$ Δ , there are three distinct sequences of numbers: the first coming from the eigenvalues of $$L^{2}$$ L 2 eigenfunctions, the second coming from resonances associated with the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $$\mathcal {Z}_-(s,z)$$ Z - ( s , z ) and $$\mathcal {Z}_+(s,z)$$ Z + ( s , z ) that encode the spectrum of $$\Delta $$ Δ in such a way that they can be used to define the regularized determinant of $$\Delta -z(1-z)I$$ Δ - z ( 1 - z ) I . The resulting formula for the regularized determinant of $$\Delta -z(1-z)I$$ Δ - z ( 1 - z ) I in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $$z\leftrightarrow 1-z$$ z ↔ 1 - z .

The structure of the extended Selberg class of degree one was completely revealed by Kaczorowski and Perelli. In this short paper, we give a new characterizationof the functions with periodic coefficients in that class by giving a simple relation that the coefficients have to satisfy.

Let $\Lambda = \{\lambda_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{\lambda_{k} \in \Lambda : | \lambda_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $\Lambda$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $\Lambda$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-\Lambda$ and the function $f$ as constructed as a Weierstrass product. In the case $f$ admits a Dirichlet series expansion in some right half-plane, we derive the meromorphic continuation in $s$ of $\Z_{f}(s,z)$ as an integral transform of $f'/f$. We apply these results to obtain superzeta product evaluations of Selberg zeta function associated to finite volume hyperbolic manifolds with cusps.

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\phi(s)$ denote the automorphic scattering determinant. From the known functional equation $\phi(s)\phi(1-s)=1$ one concludes that $\phi(1/2)^{2} = 1$. However, except for the relatively few instances when $\phi(s)$ is explicitly computable, one does not know $\phi(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $\phi(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $\phi(s)$. Then $\phi(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.

. Let E be Galois extension of Q of finite degree and let π and π ′ be two irreducible automorphic unitary cuspidal representations of GL m ( E A ) and GL m ′ ( E A ), respectively. We prove an asymptotic formula for computation of coefficients γ π,π ′ ( k ) in the Laurent (Taylor) series expansion around s = 1 of the logarithmic derivative of the Rankin-Selberg L − function L ( s,π × e π ′ ) under assumption that at least one of representations π , π ′ is self-contragredient and show that coefficients γ π,π ′ ( k ) are related to weighted Selberg orthogonality. We also replace the assumption that at least one of representations π and π ′ is self-contragredient by a weaker one.