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.

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $\zeta^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $\zeta^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants.

Abstract We define modified Li coefficients, called τ-Li coefficients for a very broad class S ♯ ♭ ( σ 0 , σ 1 ) of L-functions that contains the Selberg class, the class of all automorphic L-functions and the Rankin–Selberg L-functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class S ♯ ♭ ( σ 0 , σ 1 ) , derive an arithmetic formula for the computation of τ-Li coefficients and conduct numerical investigation of τ-Li coefficients for a certain product of shifts of the Riemann zeta function.

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weight four and six; the smallest weight cusp form Delta has weight twelve and can be written as a polynomial in E4 and E6; and the Hauptmodul j can be written as a multiple of E4 cubed divided by Delta. The goal of the present article is to seek generalizations of these results to some other genus zero arithmetic groups, namely those generated by Atkin-Lehner involutions of level N with square-free level N.

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein series. In the case of a non-compact finite volume hyperbolic Riemann surface, we finally show how to express the parabolic Eisenstein series in terms of the integral transform of a wave operator.

We develop two applications of the Kronecker’s limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil’s reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic fixed points for a few small level moonshine groups.RésuméDans cet article nous développons deux applications de la formule limite de Kronecker associée aux series d’Eisenstein elliptiques: Un théorème de factorisation pour des formes modulaires holomorphes et une preuve de la loi de réciprocité de Weil. Plusieurs exemples de notre résultat général de factorisation sont donnés, particulièrement pour quelques groupes de type moonshine, groupes de congruence et, plus généralement, pour des groupes non-cocompactes à une seule pointe. En particulier, nous calculons la fonction limite de Kronecker associée à certains points elliptiques pour des groupes de type moonshine de petit niveau.

ABSTRACT For any square-free integer N such that the “moonshine group” Γ0(N)+ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmodul of Γ0(N)+ to certain McKay–Thompson series associated to the representation theory of the Fischer–Griess monster group. In particular, the Hauptmoduli admits a q-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus groups Γ0(N)+. For all such arithmetic groups of genus up to and including three, we prove that the corresponding function field admits two generators whose q-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether i∞ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$ . Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$ , where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$ . Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$ , an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$ , or, equivalently, the zeros of $Z_{M}H_{M}$ . Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$ , and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.