Related papers: Hankel determinants of Dirichlet series

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of…

We show that the Generalized Riemann Hypothesis for all Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of $\zeta(s)$ satisfy those…

For $ t \in [0,1]$ let $\underline{H}_{2\lfloor nt \rfloor} = ( m_{i+j})_{i,j=0}^{\lfloor nt \rfloor} $ denote the Hankel matrix of order $2\lfloor nt \rfloor$ of a random vector $(m_1,\ldots ,m_{2n})$ on the moment space…

In this paper, we give an analogue of Wilton's product formula for Dirichlet series that satisfy Hecke's functional equation. We apply our results to obtain identities for Hecke series, L-functions associated to modular forms, Ramanujan's…

By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant…

We characterize Fredholm determinants of a class of Hankel composition operators via matrix-valued Riemann-Hilbert problems, for additive and multiplicative compositions. The scalar-valued kernels of the underlying integral operators are…

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of…

In this paper we determine the upper bounds of $|H_{2}(3)|$ for the inverse functions of functions of some classes of univalent functions, where $H_{2}(3)(f)=a_{3}a_{5}-a_{4}^{2}$ is the Hankel determinant of a special type.

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support.…

We introduce a sequence of orthogonal polynomials whose associated moments are the Rayleigh-type sums, involving the zeros of the Bessel derivative $J_\nu'$ of order $\nu$. We also discuss the fundamental properties of those polynomials…

Let ${\mathcal A}$ be the class of functions that are analytic in the unit disc ${\mathbb D}$, normalized such that $f(z)=z+\sum_{n=2}^\infty a_nz^n$, and let class ${\mathcal U}(\lambda)$, $0<\lambda\le1$, consists of functions…

In this short note, we compute, for large n the determinant of a class of n x n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the…

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…

We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant…

Assuming the Riemann hypothesis, we show that a certain vertical distribution of the nontrivial zeros of the Riemann zeta-function is equivalent to the generalized Riemann hypothesis for Dirichlet $L$-functions. Furthermore, under both the…

In this survey we show how to produce asymptotics of determinants of structured matrices using operator theory methods. We describe the asymptotics for finite Toeplitz matrices, finite Toeplitz plus Hankel matrices and generalizations of…

In 1975 K. Michael Day produced an exact formula for the determinants of finite Toeplitz matrices whose symbols are rational. The answer is a sum that involves powers of the roots of the numerator of the symbol and whose coefficients depend…

We consider a family of all analytic and univalent functions (i.e., one-to-one) in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we obtain the sharp bounds of the second…

We show that the ratio of a discrete Toeplitz/Hankel determinant and its continuous counterpart equals a Freholm determinant involving continuous orthogonal polynomials. This identity is used to evaluate a triple asymptotic of some discrete…