Related papers: Integers and Prime Numbers: Deriving Zipf's Law
Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or,…
By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent…
Zipf's law on word frequency is observed in English, French, Spanish, Italian, and so on, yet it does not hold for Chinese, Japanese or Korean characters. A model for writing process is proposed to explain the above difference, which takes…
The index of codivisibility of a set of integers is the size of its largest subset with a common prime divisor. For large random samples of integers, the index of codivisibility is approximately normal.
Using data from gene expression databases on various organisms and tissues, including yeast, nematodes, human normal and cancer tissues, and embryonic stem cells, we found that the abundances of expressed genes exhibit a power-law…
In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved:…
We establish the rate of convergence in the strong law of large numbers of discrete Fourier Transform of the identically distributed random variables with finite moment of order p, where 1<p<2.
In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes…
We perform a quantitative analysis of extensive chess databases and show that the frequencies of opening moves are distributed according to a power-law with an exponent that increases linearly with the game depth, whereas the pooled…
There is a probability charge on the power set of the integers that gives probability $1/p$ to every residue class modulo a prime $p$. There exists such a charge that gives probability $w$ to the set of prime numbers iff $w \in [0,1/2]$.…
We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…
Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…
This work proves that ranks and shares are statistically dependent on one another, based on simple combinatorics. It presents a formula for rank-share distribution and illustrates that Zipfs law, is descended from expected values of various…
An important body of quantitative linguistics is constituted by a series of statistical laws about language usage. Despite the importance of these linguistic laws, some of them are poorly formulated, and, more importantly, there is no…
We introduce \emph{patterned numbers}, a digit--divisor-based classification of integers motivated by recreational mathematics. A number is defined to be patterned if at least one of its positive divisors appears as a digit in its base-10…
In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
Power-law distributions with various exponents are studied. We first introduce a simple and generic model that reproduces Zipf's law. We can regard this model both as the time evolution of the population of cities and that of the asset…
A simple method to derive parametric analytical extensions of Benford's law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely the two-sided power Benford distribution and the new…
Zipf's law is the main regularity of quantitative linguistics. Despite of many works devoted to foundations of this law, it is still unclear whether it is only a statistical regularity, or it has deeper relations with information-carrying…
Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…