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The modified Bessel functions $K_{\nu}(z)$, or, for brevity, K-Bessel functions, arise at key places in analytic number theory. In particular, they appear in beautiful arithmetic identities. A survey of these arithmetical identities and…
Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also…
Basic properties of Fourier integral operators on the torus are studied by using the global representations by Fourier series instead of local representations. The results can be applied to weakly hyperbolic partial differential equations.
On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and…
This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m},…
We consider the generalized weighted zeta function for a finite digraph, and show that it has the Ihara expression, a determinant expression of graph zeta functions, with a certain specified definition for inverse arcs. A finite digraph in…
We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.
An analytic function can be continued across an analytic arc $\Gamma$ with the help of the Schwarz function $S(z)$, the analytic function satisfying $S(z) = \bar z$ for $z\in \Gamma$. We show how $S(z)$ can be computed with the AAA…
New identities on traces of representations of the Hecke algebra on the spaces of paths on graphs are presented. These identities are relevant in the computation of partition functions with fixed boundary conditions and of two-point…
An analytic Bethe ansatz is carried out related to tensor-like representations of the type II Lie superalgebras B(r|s)=osp(2r+1|2s) (r > -1, s >0) and D(r|s)=osp(2r|2s) (r >1, s >0). We present eigenvalue formulae of transfer matrices in…
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $\tau$-functions of hypergeometric type serving as…
Symbolic computation techniques are used to derive some closed form expressions for an analytic continuation of the Euler-Zagier zeta function evaluated at the negative integers as recently proposed by B. Sadaoui. This approach allows to…
This is an expository paper on the meromorphic continuation of zeta functions with Euler products (for example zeta functions of groups and height zeta functions) or without (for example the Goldbach zeta function). As an application we…
We show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this…
We give an elementary construction of representing systems of the Cauchy kernels in the Hardy spaces $H^p$, $1 \le p <\infty$, as well as of representing systems of reproducing kernels in weighted Hardy spaces.
The first step in the formulation and study of the Riemann Hypothesis is the analytic continuation of the Riemann Zeta Function (RZF) in the full Complex Plane with a pole at $s=1$. In the current work, we study the analytic continuation of…
In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of…
In this paper, we give a short elementary proof of the well known Euler's recurrence formula for the Riemann zeta function at positive even integers and integral representations of the Riemann zeta function at positive integers and at…
In this paper, we define edge zeta functions for spherical buildings associated with finite general linear groups. We derive elegant formulas for these zeta functions and reveal patterns of eigenvalues of these buildings, by introducing and…
To a representation of $\O_N$ (the Cuntz algebra with $N$ generators) we associate a projection valued measure and we study the case when this measure has atoms. The main technical tool are the spaces invariant for all the operators…