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 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 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 Proof the Skewes’ number is not an integer using lattice points and tangent line

2021 ◽  
Vol 17 (2) ◽  
pp. 5-18
Author(s):  
V. Ďuriš ◽  
T. Šumný ◽  
T. Lengyelfalusy
Keyword(s):  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Integral Function ◽  
Lattice Points ◽  
Tangent Line ◽  
Miller Test ◽  
Logarithmic Integral ◽  
The Difference ◽  
Prime Counting Function

Abstract Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

Chebysev’s Estimate of the Prime Counting Function

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Prime Number ◽  
Historical Perspective ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Counting Function

In this article we shall discuss about Chebyshev’s estimates of the prime counting function, which was later superseded by the Prime Number Theorem, nonetheless it is significant from both mathematical and historical perspective. Chebyshev’s estimates of the prime counting function forms the basis and motivation for the Prime number theorem derived later by mathematicians.

Chebysev’s Estimate of the Prime Counting Function

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Prime Number ◽  
Historical Perspective ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Prime Counting Function

In this article we shall discuss about Chebyshev’s estimates of the prime counting function, which was later superseded by the Prime Number Theorem, nonetheless it is significant from both mathematical and historical perspective. Chebyshev’s estimates of the prime counting function forms the basis and motivation for the Prime number theorem derived later by mathematicians.

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

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 On a Family of Sequences Related to Prime Counting Function

2018 ◽  
Vol 7 (6) ◽  
pp. 149
Author(s):  
Altug Alkan ◽  
Orhan Ozgur Aybar
Keyword(s):  
Initial Conditions ◽  
Counting Function ◽  
Integer Sequences ◽  
New Family ◽  
Prime Counting Function

In this study, we introduce a new family of integer sequences which are related to prime-counting function and we focus on some properties of these sequences. Sequence A316434 in OEIS is the fundamental member of solution family that we study. More precisely, we investigate the solutions of recurrence a(n) = a(π(n)) + a(n-π(n)) with some natural initial conditions where π(n) is defined by A000720 in OEIS.

scholarly journals Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

2013 ◽  
Vol 149 (7) ◽  
pp. 1129-1149 ◽  
Author(s):  
Dimitris Koukoulopoulos
Keyword(s):  
Prime Number ◽  
Elementary Proof ◽  
Counting Function ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Multiplicative Functions ◽  
Primes In Arithmetic Progressions

AbstractBuilding on the concept ofpretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

New consequences of prime-counting function

2021 ◽  
Vol 27 (4) ◽  
pp. 25-31
Author(s):  
Sadani Idir ◽  
Keyword(s):  
Counting Function ◽  
Prime Numbers ◽  
Prime Counting Function

Our objective in this paper is to study a particular set of prime numbers, namely \ left \ {p \ in \ mathbb {P} \ \ text {and} \ \ pi (p) \ notin \ mathbb {P} \ right \} \ !.As a consequence, estimations of the form \ sum {f (p)}with p being prime belonging to this set are derived.

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

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