Related papers: Digits of pi: limits to the seeming randomness II
Let $d\in\mathbb{N}$, and let $\pi$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values…
The Central Limit Theorem does not hold for strongly correlated stochastic variables, as is the case for statistical systems close to criticality. Recently, the calculation of the probability distribution function (PDF) of the magnetization…
Large Language Models (LLMs) exhibit impressive performance on complex reasoning tasks, yet they frequently fail on basic numerical problems, producing incorrect outputs. Inspired by Benford's Law, a statistical pattern in which lower…
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…
Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let \phi denote Euler's totient function, let \sigma be the sum-of-divisors…
In this paper we propose the use of $\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process $\de X_t = b(X_t, \theta)\de t + \sigma(X_t, \theta)\de W_t$, from discrete…
Benford's Law (BL) or the Significant Digit Law defines the probability distribution of the first digit of numerical values in a data sample. This Law is observed in many naturally occurring datasets. It can be seen as a measure of…
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…
We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only…
We define cyclic $U$-statistics as a variant of $U$-statistics based on variables $X_1,\dots,X_n$ that are assumed to be cyclically ordered. We also define alternating $U$-statistics where in the definition terms are summed with alternating…
We establish why li(x) outperforms x/log x as an estimate for the prime counting function pi(x). The result follows from subdividing the natural numbers into the intervals s_k :={p_k^2,..., p_{k+1}^2-1}, k>=1, each being fully sieved by the…
In this paper, we consider U-statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central…
The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n =…
The \textit{Central Limit Theorem (CLT)} is at the heart of a great deal of applied problem-solving in statistics and data science, but the theorem is silent on an important implementation issue: \textit{how much data do you need for the…
In applied probability, the normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is…
The recent technique for estimating lower bounds of the prime counting function $\pi(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s)) \geq 2$ of special values of the zeta function claims that $\pi(x)…
Let $(\xi_k, \eta_k)_{k\geq 1}$be independent identically distributed random vectors with arbitrarily dependent positive components and $T_k:=\xi_1+\ldots+\xi_{k-1}+\eta_k$for $k\in\mathbb{N}$. We call the random sequence {T_k, k=1,2...} a…
Benford's law is an empirical edict stating that the lower digits appear more often than higher ones as the first few significant digits in statistics of natural phenomena and mathematical tables. A marked proportion of such analyses is…
This article studies statistical estimation of $\pi$ based on the fact that the ratio of the volumes of a $d$-dimensional hypersphere and a $d$-dimensional hypercube is a certain function of $\pi$, and the function depends on the dimension…
The constant $\pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections…