Related papers: Notes on Universality in Short Intervals and Expon…

This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of…

New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coefficients b(k), d(k), d_(k) and d__(k). In some formal limit our expansion b(k) obtained via the alternating series gives…

We extend to several dimensions the result of K. Seip and Y.I. Lyubarskii that proves the existence of Riesz basis of exponentials for a finite union of intervals with equals lengths.

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the…

In this paper, we extend the result of Fujii on the second moment of $S(t+h) - S(t)$ to longer range of $h$ under the Riemann Hypothesis and an quantitative form of the Twin Prime Conjecture.

Time-like and space-like invariant space-time intervals are used to analyse measurements of spatial and temporal distances defined by two spatially-separated clocks. The time dilatation effect is confirmed, but not `relativity of…

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel…

Using the relativistic concept of time dilation we show that a superposition of gravitational potentials can lead to nonunitary time evolution. For sufficiently weak gravitational potentials one can still define, for all intents and…

Fujii investigated the uniform distribution of various sequences associated with the non-trivial zeros of the Riemann zeta function by evaluating certain exponential sums over these zeros. In this paper, we present analogous results for a…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x)…

We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint…

Within the framework of the Kibble-Zurek Mechanism, we investigate the universal behavior of the non-equilibrium critical fluctuations, using the Langevin dynamics of model A. With properly located typical time, length and angle scales,…

We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by…

Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…

This is a semi-expository paper on the easier aspects of the Explicit Formula for the Riemann Zeta Function. The topics reviewed here include: Weil's criterion for the Riemann Hypothesis and its probabilistic interpretation, various…

We propose the general way of study the universal estimator for the regression problem in learning theory considered in "Universal algorithms for learning theory Part I: piecewise constant functions" and "Universal algorithms for learning…

In this paper we consider generalization of classical and quantum mechanics that directly follows from the causality principle and topology of a system state space. In generalized mechanics, the Hamiltonian/Schrodinger equations remain the…

We introduce conformal transformations in the synthetic setting of metric spaces and Lorentzian (pre-)length spaces. Our main focus lies on the Lorentzian case, where, motivated by the need to extend classical notions to spaces of low…

Using the resonance method, we obtain refined estimates for joint extreme values of the Riemann zeta function at harmonic points, improving upon Levinson's 1972 results and providing new insight into the behavior of the Riemann zeta…

We give a short probabilistic (a Brownian motion) proof of the Riemann hypothesis based on some surprising, unexpected and deep algebraic conjecture (MAC in short) concerning the relation between the Riemann zeta $\xi$ and a trivial zeta…