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.

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 .

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 $\phi(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.