Related papers: An improved analytic Method for calculating $\pi(x…

This paper describes recent advances in the combinatorial method for computing $\pi(x)$, the number of primes $\leq x$. In particular, the memory usage has been reduced by a factor of $\log x$, and modifications for shared- and…

We describe a rigorous implementation of the Lagarias and Odlyzko Analytic Method to evaluate the prime counting function and its use to compute unconditionally the number of primes less than $10^{24}$.

The prime-counting function $\pi(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi(x)$ with time…

Based on the first 25 known values of Pi(10^n), the number of primes less than 10^n, with n integer between 1 and 25, we propose a conjectured value range of Pi(10^26) calculated by using polynomial interpolations with two corrective…

By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).

In this paper we discuss a method to express the Prime counting function as a "sum" over Non-trivial zeros of Riemann Zeta function, using techniques from Analytic Number Theory, also we apply our results to the sum over primes of any…

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…

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…

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.

In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as…

Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}\pi(x)=\Theta(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber…

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) +…

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.

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

We show the following bounds on the prime counting function $\pi(x)$ using principles from analytic number theory, giving an estimate: $$2 \log 2 \geq \limsup_{x \rightarrow \infty} \frac{\pi(x)}{x / \log x} \geq \liminf_{x \rightarrow…

Prime number theorem asserts that (at large $x$) the prime counting function $\pi(x)$ is approximately the logarithmic integral $\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N…

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev…

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 provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function $\psi(x)$ to bounds of the same type on both $\theta(x)$ and $\pi(x)$. This follows up our previous work on $\psi(x)$ in…

In this paper we give a new semiprimality test and we construct a new formula for $\pi ^{(2)}(N)$, the function that counts the number of semiprimes not exceeding a given number $N$. We also present new formulas to identify the $n^{th}$…