Related papers: An induction property for prime counting functions

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative…

We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-\delta}$, a Brun-Titchmarsh bound, and Linnik's…

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short…

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…

Let $\mu(n)$ denote the M\"obius function, define $M(x)= \sum_{n\leq x}^{}\mu (n)$. The main result of this paper is to prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*} which is equivalent to…

In this paper, we build some ergodic theorems involving function $\Omega$, where $\Omega(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any…

We study the asymptotic behaviour of the well-known Dykstra's algorithm through the lens of proof-theoretical techniques. We provide an elementary proof for the convergence of Dykstra's algorithm in which the standard argument is stripped…

For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

In this paper we introduce the prime index function \begin{align}\iota(n)=(-1)^{\pi(n)},\nonumber \end{align} where $\pi(n)$ is the prime counting function. We study some elementary properties and theories associated with the partial sums…

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in…

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper we consider the analogous multiplicative setting of the cyclic group $\left(\mathbb{Z}/ q\mathbb{Z}\right)^{\times}$, and…

Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results…

We prove a harmonic asymptotics density theorem for asymptotically flat initial data sets with compact boundary that satisfy the dominant energy condition. We use this to settle the spacetime positive mass theorem, with rigidity, for…

We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches…

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an…

We establish asymptotic formulae for the number of $k$-free values of polynmilas $F(x_1,\cdots,x_n)\in\mathbb{Z}[x_1,\cdots,x_n]$ of degree $d\geq 2$ for any $n\geq 1$, including when the variables are prime, as long as $k\geq (3d+1)/4$.…

Let \(u\neq \pm 1,v^2\) be a fixed integer, let \(p\geq 2\) be a prime, and let $\text{ord}_p(u) \mid p-1$ be the multiplicative order of $u \text{ mod } p$. Define a prime counting function by $\pi(u,x)=\# \{ p\leq x:\text{ord}_p(u)=p-1…

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the…

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…