Related papers: Digits of pi: limits to the seeming randomness

The occurrence of the nonzero leftmost digit, i.e., 1, 2, ..., 9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic…

Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci-…

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:…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

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 work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing…

Nature and our world have a bias! Roughly $30\%$ of the time the number $1$ occurs as the leading digit in many datasets base $10$. This phenomenon is known as Benford's law and it arrises in diverse fields such as the stock market,…

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…

We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers as bases, which leads to expansions with…

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq1}$ and $\{b_n\}_{n\geq1}$ of complex numbers whose limiting empirical measures are same. By choosing…

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function…

Researchers have observed that the frequencies of leading digits in many man-made and naturally occurring datasets follow a logarithmic curve, with digits that start with the number 1 accounting for $\sim 30\%$ of all numbers in the dataset…

This paper is concerned with the asymptotic behavior of sums of terms which are a test function f evaluated at successive increments of a discretely sampled semimartingale. Typically the test function is a power function (when the power is…

A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $\sigma$ in $\pi_n$ converges to $1/|\sigma|!$ as $n\to\infty$. We prove that $\pi_1,\pi_2,\dots$ is quasirandom if and…

We joke about how to compute (promptly) the digits of $\pi$, in base 5, from a given place without computing preceding ones.

We consider the probability distributions of values in the complex plane attained by Fourier sums of the form \sum_{j=1}^n a_j exp(-2\pi i j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If…

Given a base $b$, a "digit map" is a map $f: \mathbb{Z}^{\ge 0} \to \mathbb{Z}^{\ge 0}$ of the form $f(\sum_{i=0}^n a_ib^i) = \sum_{i=0}^n f_*(a_i)$, $0 \le a_i \le b-1$ for each $i$, where $f_* : \{0,1,\dots, b-1\} \to \mathbb{Z}^{\ge 0}$…

Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbol{\pi}}(2i-1),{\boldsymbol{\pi}}(2i))$ for a uniformly random permutation ${\boldsymbol{\pi}}$ of…

We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical…

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt{\log n/n}$. Our proofs rely on…