Related papers: A new method for computing number PI
Definition of the number of prime numbers in the given interval
We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.
We propose an automated method for proving termination of $\pi$-calculus processes, based on a reduction to termination of sequential programs: we translate a $\pi$-calculus process to a sequential program, so that the termination of the…
In this short note, we have defined a new "nested square root" function which generates usual Pi number for $x=2$. We have given some useful identities and asymptotic formulas of the Pi-function.
In an attempt to provide an answer to the increasing criticism against p-values and to bridge the gap between statistical inference and prediction modelling, we introduce the probability of improved prediction (PIP). In general, the PIP is…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
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).
I present a few new and recent ideas of the multiloop calculations.
In this paper, we present a fixed point method for high-precision computation of number $\pi$ based on the sine function. Let $P\in \mathbb{N}$. We define the function: \[ S\left(x\right) =x+\sum_{k=1}^{P}\left(\prod_{\ell=1}^{k-1}\frac…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
In this work, we obtain an iterative formula that can be used for computing digits of $\pi$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, where $c_0 = 0$ and $c_n = \sqrt{2 + c_{n - 1}}$. We also show how with the help of this…
We give a new procedure in Maple for finding the k-th power of a martix. The algorithm is based on the article [1].
New formulas for 1/Pi^2 are found by transforming Guillera's formulas
We present a new form of the Machin-like formula for $\pi$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $\pi$ due to rapidly increasing integers at each step…
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 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.
In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.
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.
This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set…
We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.