Related papers: On Hilbert-Polya conjecture: Hermitian operator na…
In this article we introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers…
We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large…
In this paper, some new results are reported for the study of Riemann zeta function $\zeta(s)$ in the critical strip $0<Re(s)<1$, such as $\zeta(s)$ expressed in a generalized Euler product only involving prime numbers. Particularly, some…
The Laplace operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions…
Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the…
Consider a family of bounded domains $\Omega_{t}$ in the plane (or more generally any Euclidean space) that depend analytically on the parameter $t$, and consider the ordinary Neumann Laplacian $\Delta_{t}$ on each of them. Then we can…
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…
We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…
We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, showing that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a semidefinite inner…
Comparison between the exact value of the spectral zeta function, $Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of…
A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\zeta_{k}$ and the method of the proof of the…
We present a proof of the functional equation of the Riemann zeta-function or more precisely the Dirichlet eta-function, which proof seems to be new but follows almost immediately from Malmst\`en's paper ``De integralibus quibusdam…
Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the…
Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value $s =…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
The $\theta=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions…
Several identities for the Riemann zeta-function $\zeta(s)$ are proved. For example, if $s = \sigma + it$ and $\sigma > 0$, then $$ \int_{-\infty}^\infty |{(1-2^{1-s})\zeta(s)\over s}|^2dt = {\pi\over\sigma}(1 -…
The Hamiltonian reggeon acting in Bargmann space is non-Hermitian with respect to the standard scalar product associated to Bargmann space. Hence the question arises, whether the eigenfunctions by the finite norm condition form a complete…
One approach to multivariate operator theory involves concepts and techniques from algebraic and complex geometry and is formulated in terms of Hilbert modules. In these notes we provide an introduction to this approach including many…
Let $\zeta^*(s)=\sum_{n=1}^{+\infty}(-1)^n/n^s$ and $\tau$ the operator defined on the Frechet space of holomorphic functions in $\{s\in \mathbb C :1/2< Re \, s<1\}$ by $\tau f(s)= f(s-2i\pi/\log 2)$. We show that the Riemann Hypothesis is…