Related papers: Gaussian Primes in Narrow Sectors

The use of Fermat-Torricelli points can be an effective mathematical tool for analyzing numerical series that have a large variance, a pronounced nonlinear trend, or do not have a normal distribution of a random variable. Linear…

We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.

We prove explicit versions of Cram\'er's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more…

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding…

Using a Zariski topology associated to a finite field extensions, we give new proofs and generalize the primitive and normal basis theorems.

We present here an elementary example, for every fixed positive integer $k,$ of a strictly stationary nongaussian stochastic process in discrete time, all of whose $k$-marginals are gaussian.

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…

Bertrand's Postulate states about the prime distribution for the real numbers. The generalization of Bertrand's Postulate was proved by Das et al. [Arxiv 2018]. In this paper, we have formalized this idea for the Gaussian primes (or the…

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a…

Random fields in nature often have, to a good approximation, Gaussian characteristics. For such fields, the relative densities of umbilical points -- topological defects which can be classified into three types -- have certain fixed values.…

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both…

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes…

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

The Gaussian theory of errors has been generalized to situations, where the Gaussian distribution and, hence, the Gaussian rules of error propagation are inadequate. The generalizations are based on Bayes' theorem and a suitable measure.…

Through a reformulation of the local limit theorem and law of small numbers, which is obtained by working in the spaces naturally associated to the limiting distributions, we discover a general and abstract framework for the investigation…

A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in…