Related papers: On a sequence involving the prime numbers
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.
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop…
An asymptotic formula is given for the number of y-smooth numbers up to x in a Beatty sequence corresponding to an irrational number of finite type.
Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…
In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed…
In this paper, we study some combinations of the degenerate and incomplete Stirling numbers of the second kind. We use a combinatorial approach and provide some asymptotic results.
We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such…
Composition methodologies in the current literature are mainly to promote estimation efficiency via direct composition, either, of initial estimators or of objective functions. In this paper, composite estimation is investigated for both…
A sharp asymptotic formula for the sum of reciprocals of $\pi(n)$ is derived, where $\pi(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivi\'c and L. Panaitopol.
We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$…
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…
We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on previously known…
Let $\pi(x;\gamma_1,\gamma_2)$ denote the number of primes $p$ with $p\leqslant x$ and $p=\lfloor n^{1/\gamma_1}_1\rfloor=\lfloor n^{1/\gamma_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in\mathbb{R}$ and…
We prove asymptotic results for the singular series associated to the distribution of three primes. Assuming a quantitative version of Hardy and Littlewood's conjecture on prime 3-tuples, we deduce an asymptotic formula related to the joint…
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we…
Several methods are available in the literature to stochastically compare random variables and random vectors. We introduce the notion of asymptotic stochastic order for random processes and define four such orders. Various properties and…