Related papers: Coordinate Distribution of Gaussian Primes
We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…
We uncover geometric aspects that underlie the sum of two independent stochastic variables when both are governed by q-Gaussian probability distributions. The pertinent discussion is given in terms of random vectors uniformly distributed on…
We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some…
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…
Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…
We consider the Gaussian limit law for the distribution of the middle prime factor of an integer, defined according to multiplicity or not. We obtain an optimal bound for the speed of convergence, thereby improving on previous estimates…
In this paper we will propose a strategy to prove Goldbach's conjecture: every even integer greater than 2 can be written as the sum of two primes.
We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen's expansion approach…
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…
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime…
The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum…
A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
We present a variety of prime-generating constructions that are based on sums of primes. The constructions come in all shapes and sizes, varying in the number of dimensions and number of generated primes. Our best result is a construction…
We give a necessary and sufficient condition for symmetric infinitely divisible distribution to have Gaussian component. The result can be applied to approximation the distribution of finite sums of random variables. Particularly, it shows…
We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of improving the current bounds on $K$. Finally, we improve the…
Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by…
Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…