Related papers: Dynamical Sieve of Eratosthenes
We review some of the recent developments and prove new results concerning frames and Bessel systems generated by iterations of the form $\{A^ng: g\in G,\, n=0,1,2,\dots \}$, where $A$ is a bounded linear operators on a separable complex…
The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…
Differential equations based on physical principals are used to represent complex dynamic systems in all fields of science and engineering. Through repeated use in both academics and industry, these equations have been shown to represent…
The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in…
We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the…
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).
It is known that prime numbers occupy specific geometrical patterns or moduli when numbers from one to infinity are distributed around polygons having sides that are integer multiple of number 6. In this paper, we will show that not only…
We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…
Some linear dynamical systems over finite fields are studied and the self-similar character of their development is proved. Connections with aperiodic tilings, Delanoy numbers and other topics are also proved. The prime fields F_p have a…
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…
Dynamical processes can be classified in various ways as deterministic or stochastic, and continuous or discrete time. All these types can be studied by the path-spaces they generate, and stationary measures on that path-space. Such…
We have been studying Eratosthenes sieve as a discrete dynamic system, obtaining exact models for the relative populations for small gaps (currently gaps $g \le 82$) in the cycle of gaps ${\mathcal G}(p^\#)$ at each stage of the sieve. The…
We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m…
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set…
This article develops a new sieve method which by adding an additional axiom to the classical formulation breaks the well-known parity problem and allows one to detect primes in thin, interesting integer sequences. In the accompanying paper…
We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity…
In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed…
Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful.…
A permutiple is a natural number whose representation in some base, $b>1$, is an integer multiple of a number whose base-$b$ representation has the same collection of digits. Previous efforts have made progress in finding such numbers using…
The prime-counting function $\pi(x)$ which returns the number of primes smaller or equal to a given number is a topic of interest in number theory. An algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called $Z$-functions, was…