Related papers: Frame Sets and Zeros of Zak transforms of Extended…
We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for…
We prove that the dynamical zeta function $Z(s)$ associated to $z^2 + c$ with $c < -3.75$ has essential zero-free strips of size $1/2 +$, that is, for every $\epsilon > 0$, there exist only finitely many zeros in the strip $\mathrm{Re}(s) >…
Our main result states that when A, B, C are subsets of Z/NZ of respective densities \alpha,\beta,\gamma, the sumset A + B + C contains an arithmetic progression of length at least e^{c(\log N)^c} for densities \alpha > (\log N)^{-2 +…
It is shown that, for every integer n>2, there exists \delta_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function \zeta_{n} contains to the interval [-\delta_{n},b^{n}].…
The singularities of the $\Gamma$ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers $n$ and $k$, $$…
This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…
Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on $\mathbb{R}_+$ such that for all $t > 0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system…
In this paper the Gaussian integral is proven using contour integration on $\frac{1}{e^{x^2}+1}$ and linking it using a limit to said Gaussian integral. The limit is alsorelated to the Riemann Zeta function using a few manipulations. This…
We characterize exponential systems on sets of finite measure that form a frame or a Riesz sequence at the critical density. Namely, they are precisely those systems for which the underlying point set admits a weak limit that yields a Riesz…
We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by the Sine$_\beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power…
The property of zeta-functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with accuracy $\epsilon>0$ by suitable vertical shifts of the pair consisting…
We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely…
It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive…
Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite…
Various mixing properties of $\beta$-, $\beta'$- and Gaussian Delaunay tessellations in $\mathbb{R}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $\beta$-mixing. In the $\beta$- and the Gaussian…
The null vectors of an arbitrary highest weight representation of the $WA_2$ algebra are constructed. Using an extension of the enveloping algebra by allowing complex powers of one of the generators, analysed by Kent for the Virasoro…
Let K be any compact set in the complex plane that has a connected complement, let A(K) be the uniforn algebra of all continuous complex functions on K that are holomorphic on the interior of K, let bK be the topological boundary of K, let…
Let $n$ be a natural number larger than two. Let $D_{2n}=\langle r,s : r^{n}=s^{2}=e, srs=r^{n-1} \rangle$ be the Dihedral group, and $\kappa $ an $n$-dimensional unitary representation of $D_{2n}$ acting in $\mathbb{C}^n$ as follows.…
It is shown that the normalized fluctuations of Riemann's zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $\gamma _{k}$ and $\gamma _{k+x},$ with $x\sim(\log k)^{\beta}$,…
We derive various interesting properties of complex equiangular cyclic frames for many pairs (n, k) using Gauss sums and number theory. We further use these results to study the random and burst errors of some special cases of complex…