Related papers: The De Bruijn-Newman constant is non-negative

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions ${\mit \Xi}(z)$ with an integral representation of the form $\int_{0}^{+\infty}du\,{\mit \Omega}(u)\,{\rm ch}(uz)$ with a real-valued…

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class…

We consider vanishing properties of exponential sums of the Liouville function $\lambda$ of the form $$ \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq…

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, B\'{a}ez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert…

Let as usual $Z(t) = \zeta(1/2+it)\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is…

Let $n\ge1$ and $B\ge2$. A real-valued function $f$ defined on the $n$-simplex $\Delta_n$ is approximately convex with respect to $\Delta_{B-1}$ iff f(\sum_{i=1}^B t_ix_i) \le \sum_{i=1}^B t_if(x_i) +1 for all $x_1,...,x_B \in \Delta_n$ and…

In this paper we provide a proof of the Riemann Hypothesis by relating the non-trivial zeros of the zeta function to a certain Sturm-Liouville eigenvalue problem on a finite interval.

Riemann function $\xi(s)=u+iv, s=\beta+1/2+it$ has the important symmetry: $v=0$ if $\beta=0$. For $\beta>0$ we prove $|u|>0$ inside any root-interval $I_j=[t_j,t_{j+1}]$ and $v$ has opposite signs at two end-points of $I_j$. They imply…

It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=\sigma+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n…

A strategy for proving (not a proof of, as was the first over-optimistic belief) the Riemann hypothesis is suggested. The vanishing of Riemann Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D^+…

We establish the H\"older continuity of bounded nonnegative weak solutions to \begin{align*} \big(\Phi^{-1}(w)\big)_t=\Delta w+\nabla\cdot\big(a(x,t)\Phi^{-1}(w)\big)+b\big(x,t,\Phi^{-1}(w)\big), \end{align*} with convex $\Phi\in…

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the…

Given a continuous Hamiltonian $H : (x,p,u) \mapsto H(x,p,u)$ defined on $ T^*M \times \mathbb R $, where $M$ is a closed connected manifold, we study viscosity solutions, $u_\lambda : M\to \mathbb R$, of discounted equations: $ H(x, d_x…

This paper studies the local spacings of deformations of the Riemann zeta function under certain averaging and differencing operations. For real h it considers A_h(s)= 1/2(xi(s+h)+ xi(s-h)) and B_h(s)=1/(2i)(xi(s+h)-xi(s-h)), where xi(s) is…

Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(\Sigma,g)$. Precisely, if $\lambda_1(\Sigma)$ is the first eigenvalue of the…

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian $\Delta_p$, which allows us to analyse and classify non-variational eigenvalues. We show the stability of…

In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any…

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the…

We construct certain entire function $\lambda(s)$ which for integer s coincides with the well-known Keiper-Li coefficients, i.e. $\lambda(n)={\lambda}_{n}$. This is an even function ${\lambda}(s)={\lambda}(-s)$ and has an infinitude of…

Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on $\mathbb{R}$, where $\gamma>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is,…