Related papers: On the counting function of semiprimes
We prove several asymptotic continued fraction expansions of $\pi(x)$, $\Pi(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $\pi(x)$ is the prime counting function, $\Pi(x) = \sum_{k = 1}^\infty…
The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…
Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length…
In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and…
A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…
In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in…
Let $p$ be a prime number, $k\ge 0$ and $f$ be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function $$\psi_{f,k}(x):=\sum_{n\le x}\Lambda…
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a subset $S…
In this paper we review recently established results on the asymptotic behaviour of the trigonometric product $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ as $n\to \infty$. We focus on irrationals $\alpha$ whose continued fraction…
The main objective of this article is to study the asymptotic behavior of Salie sums over arithmetic progressions. We deduce from our asymptotic formula that Salie sums possess a bias of being positive. The method we use is based on…
We study certain resonance-counting functions for potential scattering on infinite cylinders or half-cylinders. Under certain conditions on the potential, we obtain asymptotics of the counting functions, with an explicit formula for the…
Motivated by real-world machine learning applications, we analyze approximations to the non-asymptotic fundamental limits of statistical classification. In the binary version of this problem, given two training sequences generated according…
The leading asymptotic behaviour of the Humbert functions $\Phi_2$, $\Phi_3$, $\Xi_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these…
One can find lists of whole numbers having equal sum and product. We call such a creature a bioperational multiset. No one seems to have seriously studied them in areas outside whole numbers such as the rationals, Gaussian integers, or…
We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…
This paper is devoted to study some expressions of the type $\prod_{p} p^{\lfloor\frac{x}{f(p)}\rfloor}$, where $x$ is a nonnegative real number, $f$ is an arithmetic function satisfying some conditions, and the product is over the primes…
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of…
We present the formalization of Dirichlet's theorem on the infinitude of primes in arithmetic progressions, and Selberg's elementary proof of the prime number theorem, which asserts that the number $\pi(x)$ of primes less than $x$ is…