scholarly journals Asymptotic counting theorems for primitive juggling patterns

2019 ◽  
Vol 15 (05) ◽  
pp. 1037-1050
Author(s):  
Erik R. Tou
Keyword(s):  
Prime Number ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Mathematical Theory ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Theoretical Framework ◽  
Positive Integers ◽  
Infinite Set ◽  
Fixed Period

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

scholarly journals Integers free of small prime factors in arithmetic progressions

2000 ◽  
Vol 157 ◽  
pp. 103-127 ◽  
Author(s):  
Ti Zuo Xuan
Keyword(s):  
Prime Number ◽  
Short Range ◽  
Error Term ◽  
Real Zero ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Real Character ◽  
Dickman Function ◽  
Prime Factors ◽  
Positive Integers

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

2019 ◽  
Vol 101 (1) ◽  
pp. 35-39 ◽  
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Positive Integers

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

scholarly journals Combinatorial Models of the Distribution of Prime Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1224
Author(s):  
Vito Barbarani
Keyword(s):  
Prime Number ◽  
Counting Function ◽  
Stirling Numbers ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Lower And Upper Bounds ◽  
Integer Sequences ◽  
New Class ◽  
Random Integer ◽  
Prime Counting Function

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

scholarly journals The mathematical constraint of the counterexample of Goldbach's strong conjecture

2020 ◽  
Author(s):  
ahmad hazaymeh ◽  
Khaled Hazaymeh ◽  
Sukaina Hazaymeh
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Counter Example ◽  
Logical Conclusion

In this paper, we have demonstrated a proof that the Counterexample of Goldbach's strong conjecture is impossible in two steps: First, we reformulated Goldbach's strong conjecture using the subtraction connotation. Second: the mathematical Constraint that must be fulfilled in any even number has been deduced to be that even number a Counterexample of Goldbach's strong conjecture. Then we demonstrated that any counterexample would fulfill this mathematical Constraint. It will either contradict the theorem of infinite prime numbers or contradict the Prime Number Theorem. Therefore, the logical conclusion is that there is no counterexample to Goldbach's strong conjecture. With the absence of a counter-example, Goldbach's strong conjecture would be a true conjecture

scholarly journals Combinatorial Models of the Distribution of Prime Numbers

2021 ◽  
Author(s):  
Vito Barbarani
Keyword(s):  
Prime Number ◽  
Counting Function ◽  
Stirling Numbers ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Lower And Upper Bounds ◽  
Integer Sequences ◽  
New Class ◽  
Random Integer ◽  
Prime Counting Function

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function &pi;(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of &pi;(x) are found.

scholarly journals An Inductive Proof of Bertrand's Postulate

2019 ◽  
Vol 38 ◽  
pp. 85-87
Author(s):  
Bijoy Rahman Arif
Keyword(s):  
Prime Number ◽  
Gamma Function ◽  
Standard Form ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Binomial Coefficients ◽  
Number Distribution ◽  
Bertrand’S Postulate ◽  
Stated Problem ◽  
Chebyshev Functions

In this paper, we are going to prove a famous problem concerning the prime numbers called Bertrand's postulate. It states that there is always at least one prime, p between n and 2n, means, there exists n < p < 2n where n > 1. It is not a newer theorem to be proven. It was first conjectured by Joseph Bertrand in 1845. He did not find a proof of this problem but made important numerical evidence for the large values of n. Eventually, it was successfully proven by Pafnuty Chebyshev in 1852. That is why it is also called Bertrand-Chebyshev theorem. Though it does not give very strong idea about the prime distribution like Prime Number Theorem (PNT) does, the beauty of Bertrand's postulate lies on its simple yet elegant definition. Historically, Bertrand's postulate is also very important. After Euclid's proof that there are infinite prime numbers, there was no significant development in the prime number distribution. Peter Dirichlet stated the standard form of Prime Number Theorem (PNT) in 1838 but it was merely a conjecture that time and beyond the scope of proof to the then mathematicians. Bertrand's postulate was a simply stated problem but powerful enough, easy to prove and could lead many more strong assumptions about the prime number distribution. Illustrious Indian mathematician, Srinivasa Ramanujan gave a shorter but elegant proof using the concept of Chebyshev functions of prime, υ(x), Ψ(x)and Gamma function, Γ(x) in 1919 which led to the concept of Ramanujan Prime. Later Paul Erdős published another proof using the concept of Primorial function, p# in 1932. The elegance of our proof lies on not using Gamma function yet finding the better approximations of Chebyshev functions of prime. The proof technique is very similar the way Ramanujan proved it but instead of using the Stirling's approximation to the binomial coefficients, we are proving similar results using well-known proving technique the mathematical induction and they lead to somewhat stronger than Ramanujan's approximation of Chebyshev functions of prime. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 85-87

The function field abstract prime number theorem

1989 ◽  
Vol 106 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Function Field ◽  
Prime Number Theorem ◽  
Arithmetical Semigroup ◽  
Enumeration Problems ◽  
Ring Of Polynomials ◽  
Arithmetical Semigroups ◽  
Positive Integers ◽  
Free Commutative Semigroup

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties(i) ∂(l) = 0,∂(p) > 0 for p∈P;(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)

The first-digit frequencies of prime numbers and Riemann zeta zeros

2009 ◽  
Vol 465 (2107) ◽  
pp. 2197-2216 ◽  
Author(s):  
Bartolo Luque ◽  
Lucas Lacasa
Keyword(s):  
Prime Number ◽  
Final Analysis ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Number Sequence ◽  
Riemann Zeta ◽  
Global Patterns ◽  
Digit Frequencies ◽  
Ultimate Nature ◽  
Shed Light

Prime numbers seem to be distributed among the natural numbers with no law other than that of chance; however, their global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes. In this paper, we show that a generalization of the well-known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the non-trivial Riemann zeta zeros. We prove that the prime number theorem is, in the final analysis, responsible for these patterns.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

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