scholarly journals An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 775
Author(s):  
Bruno Aiazzi ◽  
Stefano Baronti ◽  
Leonardo Santurri ◽  
Massimo Selva
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Integral Function ◽  
Point Of View ◽  
Binary Sequences ◽  
Prime Characteristic ◽  
Twin Primes ◽  
Logarithmic Integral ◽  
The One ◽  
Prime Function

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the case of twin primes is formulated. The proposed investigation, which is named Limited INtervals into PEriodical Sequences (LINPES) relies on a set of binary periodical sequences that are evaluated in limited intervals of the prime characteristic function. These sequences are built by considering the ensemble of deleted (that is, 0) and undeleted (that is, 1) integers in a modified version of the Sieve procedure, in such a way a symmetric succession of runs of zeroes is found in correspondence of the gaps between the undeleted integers in each period. Such a formulation is able to estimate the prime number function in an equivalent way to the logarithmic integral function Li(x). The present analysis is then extended to the twin primes, by taking into account only the runs whose size is two. In this case, the proposed procedure gives an estimation of the twin prime function that is equivalent to the one of the logarithmic integral function Li 2 ( x ) . As a consequence, a possibility is investigated in order to count the twin primes in the same intervals found for the primes. Being that the bounds of these intervals are given by squares of primes, if such an inference were actually proved, then the twin primes could be estimated up to infinity, by strengthening the conjecture of their never-ending.

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.

scholarly journals ON LITTLEWOOD’S PROOF OF THE PRIME NUMBER THEOREM

2019 ◽  
Vol 101 (2) ◽  
pp. 226-232
Author(s):  
ALEKSANDER SIMONIČ
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Prime Number Theorem ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
The One ◽  
Riemann’S Zeta Function

In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

scholarly journals Numerical analysis approach to twin primes conjecture

2021 ◽  
Vol 27 (3) ◽  
pp. 175-183
Author(s):  
Gabriele Di Pietro
Keyword(s):  
Numerical Analysis ◽  
Integral Function ◽  
Future Research ◽  
Distribution Of Primes ◽  
Analysis Method ◽  
Basic Form ◽  
Numerical Analysis Method ◽  
Twin Primes ◽  
Starting Point ◽  
Logarithmic Integral

The purpose of this paper is to demonstrate how the modified Sieve of Eratosthenes is used to have an approach to twin prime conjecture. If the Sieve is used in its basic form, it does not produce anything new. If it is used through the numerical analysis method explained in this paper, we obtain a specific counting of twin primes. This counting is based on the false assumption that distribution of primes follows punctually the Logarithmic Integral function denoted as Li(x) (see [5] and [10], pp. 174–176). It may be a starting point for future research based on this numerical analysis method technique that can be used in other mathematical branches.

scholarly journals Products of two proportional primes

2017 ◽  
Vol 13 (10) ◽  
pp. 2583-2596 ◽  
Author(s):  
Pieter Moree ◽  
Sumaia Saad Eddin
Keyword(s):  
Asymptotic Formula ◽  
Real Number ◽  
Prime Number ◽  
Prime Number Theorem ◽  
Strong Form ◽  
Number Formula ◽  
Logarithmic Integral ◽  
Rsa Cryptography

In RSA cryptography numbers of the form [Formula: see text], with [Formula: see text] and [Formula: see text] two distinct proportional primes play an important role. For a fixed real number [Formula: see text] we formalize this by saying that an integer [Formula: see text] is an RSA-integer if [Formula: see text] and [Formula: see text] are primes satisfying [Formula: see text]. Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form [Formula: see text] up to [Formula: see text], with [Formula: see text] both prime, satisfy [Formula: see text]. In this paper, we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem with the logarithmic integral replaced by a variant. From this we derive an asymptotic formula for the number of RSA-integers [Formula: see text] which is much more precise than an earlier one derived by Decker and Moree in 2008.

Understanding functional illiteracy from a policy, adult education, and cognition point of view: Towards a joint referent framework

2019 ◽  
Vol 30 (2) ◽  
pp. 109-122
Author(s):  
Aleksandar Bulajić ◽  
Miomir Despotović ◽  
Thomas Lachmann
Keyword(s):  
Adult Education ◽  
Operational Definition ◽  
Literacy Skills ◽  
Point Of View ◽  
Industrialized Countries ◽  
Functional Illiteracy ◽  
Scale Levels ◽  
Components Of Reading ◽  
The One ◽  
Definition Of

Abstract. The article discusses the emergence of a functional literacy construct and the rediscovery of illiteracy in industrialized countries during the second half of the 20th century. It offers a short explanation of how the construct evolved over time. In addition, it explores how functional (il)literacy is conceived differently by research discourses of cognitive and neural studies, on the one hand, and by prescriptive and normative international policy documents and adult education, on the other hand. Furthermore, it analyses how literacy skills surveys such as the Level One Study (leo.) or the PIAAC may help to bridge the gap between cognitive and more practical and educational approaches to literacy, the goal being to place the functional illiteracy (FI) construct within its existing scale levels. It also sheds more light on the way in which FI can be perceived in terms of different cognitive processes and underlying components of reading. By building on the previous work of other authors and previous definitions, the article brings together different views of FI and offers a perspective for a needed operational definition of the concept, which would be an appropriate reference point for future educational, political, and scientific utilization.

Recommendations of The Ombudsman’s Dilemma: Between Law Enforcement Officials and Protection of Credibility

2018 ◽  
Vol 2 ◽  
pp. 1-12
Author(s):  
Dyah Adriantini Sintha Dewi
Keyword(s):  
Law Enforcement ◽  
Public Service ◽  
Public Services ◽  
Point Of View ◽  
Law Enforcement Officials ◽  
Muslim Majority ◽  
The Republic ◽  
The Government ◽  
The One ◽  
Oversight Body

The Ombudsman as an external oversight body for official performance, in Fikih Siyasah (constitutionality in Islam) is included in the supervision stipulated in legislation (al-musahabah al-qomariyah). Supervision is done so that public service delivery to the community is in accordance with the rights of the community. This is done because in carrying out its duties, officials are very likely to conduct mal administration, which is bad public services that cause harm to the community. The Ombudsman is an institution authorized to resolve the mal administration issue, in which one of its products is by issuing a recommendation. Although Law No. 37 of 2018 on the Ombudsman of the Republic of Indonesia states that the recommendation is mandatory, theombudsman's recommendations have not been implemented. This is due to differences in point of view, ie on the one hand in the context of law enforcement, but on the other hand the implementation of the recommendation is considered as a means of opening the disgrace of officials. Recommendations are the last alternative of Ombudsman's efforts to resolve the mal administration case, given that a win-win solution is the goal, then mediation becomes the main effort. This is in accordance with the condition of the Muslim majority of Indonesian nation and prioritizes deliberation in resolving dispute. Therefore, it is necessary to educate the community and officials related to the implementation of the Ombudsman's recommendations in order to provide good public services for the community, which is the obligation of the government.

Defending the Difference

2017 ◽  
Keyword(s):  
Social Sciences ◽  
Natural Sciences ◽  
Point Of View ◽  
True Nature ◽  
Natural Entities ◽  
The Social ◽  
The World ◽  
Methodological Difference ◽  
The Difference ◽  
The One

Dreyfus argues that there is a basic methodological difference between the natural sciences and the social sciences, a difference that derives from the different goals and practices of each. He goes on to argue that being a realist about natural entities is compatible with pluralism or, as he calls it, “plural realism.” If intelligibility is always grounded in our practices, Dreyfus points out, then there is no point of view from which one can ask about or provide an answer to the one true nature of ultimate reality. But that is consistent with believing that the natural sciences can still reveal the way the world is independent of our theories and practices.

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

scholarly journals Linkages between Energy Delivery and Economic Growth from the Point of View of Sustainable Development and Seaports

Energies ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 4255
Author(s):  
Elżbieta Szaruga ◽  
Zuzanna Kłos-Adamkiewicz ◽  
Agnieszka Gozdek ◽  
Elżbieta Załoga
Keyword(s):  
Sustainable Development ◽  
Crude Oil ◽  
Cross Correlation ◽  
Point Of View ◽  
Oil Products ◽  
Modern Approach ◽  
Economic Cycles ◽  
Study Results ◽  
The One ◽  
Oil And Oil Products

This paper presents the synchronisation of economic cycles of GDP and crude oil and oil products cargo volumes in major Polish seaports. On the one hand, this issue fits into the concept of sustainable development including decoupling; on the other hand, the synchronisation may be an early warning tool. Crude oil and oil products cargo volumes are a specific barometer that predicts the next economic cycle, especially as they are primary sources of energy production. The research study applies a number of TRAMO/SEATS methods, the Hodrick–Prescott filter, spectral analysis, correlation and cross-correlation function. Noteworthy is the modern approach of using synchronisation of economic cycles as a tool, which was described in the paper. According to the study results, the cyclical components of the cargo traffic and GDP were affected by the leakage of other short-term cycles. However, based on the cross-correlation, it was proved that changes in crude oil and oil products cargo volumes preceded changes in GDP by 1–3 quarters, which may be valuable information for decision-makers and economic development planners.

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