Related papers: New bounds for the prime counting function \pi(x)
In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime…
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 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…
In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.
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…
In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates…
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…
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…
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…
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…
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 article we provide new explicit Chebyshev's bounds for the prime counting function $\psi(x)$. The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new…
In this paper, a new formula for {\pi}^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two…
We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
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…
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.
Definition of the number of prime numbers in the given interval
We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these…
The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by proving the well-known explicit Schoenfeld bound…