Let F be a number field of a finite degree and let L(s,π ×π′) be the Rankin–Selberg L-function associated to unitary cuspidal automorphic representations π and π′ of GLm(𝔸F) and GLm′(𝔸F), respectively. The main result of the paper is an asymptotic formula for evaluation of coefficients appearing in the Laurent (Taylor) series expansion of the logarithmic derivative of the function L(s,π ×π′) at s = 1. As a corollary, we derive orthogonality and weighted orthogonality relations.

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.

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.

The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.

Investment into people through education, professional training in accordance to bank’s needs, and work on “nurturing” the readiness to challenges and changes are all significant starting points for achieving bank competitiveness. Identifying the dominant forms of employee training and determining the level of employee satisfaction by the existing training programs, which is the basic goal of this paper, enable bank managers to obtain valid information on appropriate changes of certain training programs and development of employees. The paper presents the results of the empirical research conducted in a subject BiH bank, aimed at defining the methods of employee training that are characteristic of the banking sector and the level of employee satisfaction by training programs they attend. The research included 172 employees of the subject bank. The authors believe that the critical analysis of the employee training and development methods applied in the banking sector as well as the criteria for selecting the programs for the realization of these methods can lead to widening the scientific knowledge in the field of human resource management in banking and to creating specific recommendations for bank managers which they can/need apply in their practice in order to improve the entire business operations.

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.