Related papers: Zeta-Functions for Non-Minimal Operators

In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the…

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $\Delta^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero…

Some zeta functions which are naturally attached to the locally homogeneous vector bundles over compact locally symmetric spaces of rank one are investigated. We prove that such functions can be expressed in terms of entire functions whose…

An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that…

In this paper we analyze the spectral zeta function associated with a Laplace operator acting on scalar functions on an N-dimensional Euclidean space in the presence of a spherically symmetric background potential. The obtained analytic…

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…

Based on work of Alain Connes, I have constructed a spectral interpretation for zeros of L-functions. Here we specialise this construction to the Riemann zeta function. We construct an operator on a nuclear Frechet space whose spectrum is…

Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical…

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism…

We postulate the existence of a self-adjoint operator associated to a system with countably infinite number of degrees of freedom whose spectrum is the sequence of the nontrivial zeros of the Riemann zeta function. We assume that it…

This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The…

We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cram\'er, and also new representations for some arithmetical…

Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perron's formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Euler's…

Levinson and Montgomery proved that the Riemann zeta-function $\zeta(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $\zeta'(1/2+it)=0$ implies $\zeta(1/2+it)=0$.…

In this article, we introduce a general approach for deriving zero-free half-planes for the Riemann zeta function $\zeta$ by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to…

We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments $\zeta_r(s,s,\ldots,s)$ on the real line. Our basic tool is an "infinite'' version of Newton's classical identities. We carry out numerical…

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing…

In this paper, we investigate the sharing values problem that entire function $f(z)$ and its first order difference operator $\Delta_{\eta}f(z)$ share two distinct pairs of finite values IM. We prove: Let $f(z)$ be a non-constant entire…

We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix…

We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector…