Related papers: Digits of pi: limits to the seeming randomness II
In this paper, we will see that the proportion of d as leading digit, d $\in$ 1, 9, in data (obtained thanks to the hereunder developed model) is more likely to follow a law whose probability distribution is determined by a specific upper…
The following work is written in easy language for college level students. It shows how the first digit probabilities of a group of continuous real-valued functions can be calculated. Thus, examples explaining how the probabilities are…
This paper addresses the behavior of the Lov\'asz number for dense random circulant graphs. The Lov\'asz number is a well-known semidefinite programming upper bound on the independence number. Circulant graphs, an example of a Cayley graph,…
It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be…
The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form $\log(1+{1}/{d})$, where $d=1, 2, ..., 9$. Such…
Let $\mathbf{A}=\frac{1}{\sqrt{np}}(\mathbf{X}^T\mathbf{X}-p\mathbf {I}_n)$ where $\mathbf{X}$ is a $p\times n$ matrix, consisting of independent and identically distributed (i.i.d.) real random variables $X_{ij}$ with mean zero and…
Catalan's constant and the lemniscate constants have been important mathematical constants of interest to the mathematical society, yet various properties are unknown. An important property of significant mathematical constants is whether…
Let $L(s, \chi_1), \ldots, L(s, \chi_N)$ be primitive Dirichlet $L$-functions different from the Riemann zeta function. Under suitable hypotheses we prove that any linear combination $a_1\log|L(\rho,\chi_1)|+\dots+a_N\log|L(\rho,\chi_N)|$…
The Newcomb-Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset, so that, for example, the first significant digit has a probability of $30.1$ % of being $1$ and…
The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of…
Fix a positive integer $N\geq2$. For a real number $x\in[0,1]$ and a digit $i\in\{0, 1,...,N-1\}$, let $\Pi_i(x, n)$ denote the frequency of the digit $i$ among the first $n$ $N$-adic digits of $x$. It is well-known that for a typical (in…
Given natural parameters s and r, where $2\leq s\leq r$, we consider the distribution of a random variable $\xi=\sum\limits_{k=1}^{\infty}s^{-k}\xi_k\equiv\Delta^{r_s}_{\xi_1\xi_2...\xi_k...},$ where $(\xi_k)$ is a sequence of independent…
In this paper, we show the central limit theorem for the logarithmic determinant of the sample correlation matrix $\mathbf{R}$ constructed from the $(p\times n)$-dimensional data matrix $\mathbf{X}$ containing independent and identically…
We present a new record on computing specific bits of Pi, the mathematical constant, and discuss performing such computations on Apache Hadoop clusters. The specific bits represented in hexadecimal are 0E6C1294 AED40403 F56D2D76 4026265B…
A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the…
An investigation of the comparative efficiency of the different methods in which {\pi} is cal- culated. This thesis will compare and contrast five different methods in calculating {\pi} by first deriving the various proofs to each method…
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…
By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions:…
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 work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally…