Related papers: An algorithm for the prime-counting function of pr…
We analyze algorithms for computing the $n$th prime $p_n$ and establish asymptotic bounds for several approaches. Using existing results on the complexity of evaluating the prime-counting function $\pi(x)$, we show that the binary search…
The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.
Let pi(x) denote the number of primes smaller or equal to x. We compare sqrt{pi}(x) with sqrt{R}(x) and sqrt{li}(x), where R(x) and li(x) are the Riemann function and the logarithmic integral, respectively. We show a regularity in the…
Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $\pi(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that…
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.
All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of…
In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x…
For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. For integers $a$ and $m>0$, we determine when there is an integer $n>1$ with $\pi(n)=(n+a)/m$. In particular, we show that for any integers $m>2$ and $a\le\lceil…
We will generalize the combinatorial algorithms for computing $\pi(x)$ to compute sums ${F(x) = \sum_{p \leq x} p^k}$ for $k \in \mathbb{Z}_{\geq 0}$. The detailed exposition of algorithms is included along with implementation details.
In this paper we propose an algorithm that correctly discards a set of numbers (from a previously defined sieve) with an interval of integers. Leopoldo's Theorem states that the remaining integer numbers will generate and count the complete…
In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $\pi(x)$, which improve the current best estimates for large values of $x$. As an…
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…
The theorem below gives another way of computing the distribution prime counting function without using recursion and the values of Prime numbers
As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of…
The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…
An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…
We reveal a relationship between the prime counting function and an operation performed on a unique subsequence of the primes.
We note that the inequalities $0.92 \frac{x}{\log(x)} <\pi(x)< 1.11 \frac{x}{\log(x)}$ do not hold for all $x\ge 30$, contrary to some references. These estimates on $\pi(x)$ came up recently in papers on algebraic number theory.
A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function $\pi_2(x)$, the number of semiprimes less than or equal to $x$, is studied. Using a combinatorial argument,…
By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all…