Related papers: An Application of Hausdorff Moment Problem
We prove lower bounds for the discrete negative $2k$th moment of the derivative of the Riemann zeta function for all fractional $k\geqslant 0$. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general…
We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…
We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of…
We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and…
The article provides a sufficient condition for a locally finite module over the absolute Galois group of a finite field F to satisfy the Riemann Hypothesis Analogue with respect to the projective line. The condition holds for all smooth…
In this present investigation, we found a set of sufficient conditions to be imposed on the parameters of the Fox H-functions which allow us to conclude that it is non-negative. As applications, various new facts regarding the Fox-Wright…
We consider a class of abstract second order evolution equations with a restoring force that is strictly superlinear at infinity with respect to the position, and a dissipation mechanism that is strictly superlinear at infinity with respect…
In this article, we studied the inverse Erd\H{o}s-Heilbronn problem with the restricted sumset from two components $A$ and $B$ that are not necessarily the same. We give a completely elementary proof for the problem in $\mathbb{Z}$ and some…
In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenb\"ock-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar…
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes…
If a bulk gravitational path integral can be identified with an average of partition functions over an ensemble of boundary quantum theories, then a corresponding moment problem can be solved. We review existence and uniqueness criteria for…
The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…
We give necessary and sufficient conditions for differentiating under the integral sign an integral that depends on a parameter. The conditions require the equality of two iterated integrals and depend on being able to integrate every…
In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< \alpha \leq 1$ and the Riemann- Liouville fractional integral of order $\beta >0$ under certain constraints. A…
We generalize a classical theorem of Besicovitch, showing that, for any positive integers $k<n$, if $E\subset \mathbb R^n$ is a Souslin set which is not $\mathcal{H}^k$-$\sigma$-finite, then $E$ contains a purely unrectifiable closed set…
We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schr\"odinger operator $- \Delta + V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(\Omega)$. More precisely, if $u…
For a degree 2n finite sequence of real numbers $\beta \equiv \beta^{(2n)}= \{ \beta_{00},\beta_{10}, \beta_{01},\cdots, \beta_{2n,0}, \beta_{2n-1,1},\cdots, \beta_{1,2n-1},\beta_{0,2n} \}$ to have a representing measure $\mu $, it is…
We give a necessary and sufficient condition for an inverse sequence $S_0 \leftarrow S_1 \leftarrow \dots$ indexed by natural numbers to have ${\rm lim}^1S=0$. This condition can be treated as a transfinite version of the Mittag-Leffler…
To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann…
Good's Theorem for regular continued fraction states that the set of real numbers $[a_0;a_1,a_2,\ldots]$ such that $\displaystyle\lim_{n\to\infty} a_n=\infty$ has Hausdorff dimension $\tfrac{1}{2}$. We show an analogous result for the…