Related papers: Infinite order linear difference equation satisfie…

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of…

We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of…

Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log…

In this article we define and study a zeta function $\zeta_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on…

The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other…

In this note we will discuss Euler's solution of the simple difference equation that he gave in his paper{\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination…

This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical…

We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its…

Infinite order differential operators appear in different fields of Mathematics and Physics and in the last decades they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for…

We study the existence of $T$-periodic solutions $(T > 0)$ for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index…

This research expository article contains a survey of earlier work (in \S2--\S4) but also contains a main new result (in \S5), which we first describe. Given $c \geq 0$, the spectral operator $\mathfrak{a} = \mathfrak{a}_c$ can be thought…

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…

In this article, we present a solution to the problem: "Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator $-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension…

In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals,…

We consider the indefinite Sturm-Liouville differential expression \[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where $\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in I$ and the…

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$, $$ \cL^{(\alpha)}_{\sigma,b} f(x):=\mbox{p.v.} \int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\dif z+b(x)\cdot\nabla f(x)+\sL f(x), $$ where…

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute…

In the previous two parts of this series of papers, we introduced and studied a large class of analytic difference operators admitting reflectionless eigenfunctions, focusing on algebraic and function-theoretic features in the first part,…

We prove that if $\delta>0$ and $p$ is real then $$ \sup_T \int_T^{T+\delta} |\zeta(1+it)|^p dt <\infty,$$ if and only if $-1<p<1$. Furthermore, we show the omega estimates $$ \int_T^{T+\delta} |\zeta(1+it)|^{\pm 1} dt = \Omega(\log \log…

We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…