Related papers: Gaussian primes in almost all narrow sectors
We prove that the prime ideals in every class of a number field contain arbitrary large truncated ideal classes.
The Gaussian Moat problem asks whether one can walk to infinity in the Gaussian integers using the Gaussian primes as stepping stones and taking bounded length steps or not. In this paper, we have analyzed the Gaussian primes and also…
Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to…
The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp…
For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value.…
Let $K$ be a finite Galois extension of $\mathbb{Q}$. We count primes in short intervals represented by the norm of a prime ideal of $K$ satisfying a small sector condition determined by Hecke characters. We also show that such primes are…
Gaussian process is a theoretically appealing model for nonparametric analysis, but its computational cumbersomeness hinders its use in large scale and the existing reduced-rank solutions are usually heuristic. In this work, we propose a…
We collect various facts related loosely to random Gaussian quadrilaterals in the plane. For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.
Based on Euclid's algorithm, we find a kind of special sequences which play an interesting role in the study of primes. We call them W Sequences. They not only ties up the distribution of primes in short interval but also enables us to give…
Gaussian Processes (GPs) are Bayesian models that provide uncertainty estimates associated to the predictions made. They are also very flexible due to their non-parametric nature. Nevertheless, GPs suffer from poor scalability as the number…
We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high…
We show that the number of lattice points lying in a thin annulus has a Gaussian value distribution if the width of the annulus tends to zero sufficiently slowly as we increase the inner radius.
The set of minimal primes of a ring is a very important set as far the spectrum of a ring is concerned as every prime contains a minimal prime. So, knowing the minimal primes is the first (important and difficult) step in describing the…
A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…
Lower bounds for persistence probabilities of stationary Gaussian processes in discrete time are obtained under various conditions on the spectral measure of the process. Examples are given to show that the persistence probability can decay…
We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and…
In this note, we define a Gaussian probability distribution over matrices. We prove some useful properties of this distribution, namely, the fact that marginalization, conditioning, and affine transformations preserve the matrix Gaussian…
Gaussian processes (GPs) offer a principled probabilistic model over functions, but exact inference is restricted to the linear-Gaussian regime. We establish an explicit equivalence between GPs and a class of linear diffusion models,…
Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the…
We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of…