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, 2023
The counting functions of prime pairs are derived. The asymptotic behavior of the prime pair counting functions are also analyzed.
Ding, Keyang
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Chebysev’s Estimate of the Prime Counting Function

, 2020
In this article we shall discuss about Chebyshev’s estimates of the prime counting function, which was later superseded by the Prime Number Theorem, nonetheless it is significant from both mathematical and historical perspective.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 ...
Sourangshu Ghosh
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Prime Counting Function Identity

, 2022
In this paper, a new formula for π^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two primes in the product may equal each other. Semi-prime numbers are also a case of almost primes.
Garg, Suyash
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On Prime Counting Functions Using Odd $K$-Almost Primes [PDF]

European Journal of Pure and Applied Mathematics
This work takes an interesting diversion, revealing the extraordinary capacity to determine the precise number of primes in a space tripled over another. Exploring the domain of K-almost prime numbers, this paper provides a clear explanation of the complex idea.
T. Rashid +3 more
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The Prime Counting Function [PDF]

Journal of the Korea Society of Computer and Information, 2011
The Riemann`s zeta function has been known as answer for a number of primes (x) less than given number x. In prime number theorem, there are another approximation function ,Li(x), and R(x). The error about (x) is R(x) . The logarithmic integral function is Li(x)
Sang-Un Lee, Myeong-Bok Choi
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A Direct Proof of the Prime Number Theorem using Riemann's Prime-counting Function [PDF]

, 2021
In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev functions as ...
Liu, Zihao, Zihao Liu
core +1 more source

Orbit Growth of Periodic-Finite-Type Shifts via Artin–Mazur Zeta Function

Mathematics, 2020
The prime orbit and Mertens’ orbit counting functions describe the growth of closed orbits in a discrete dynamical system in a certain way. In this paper, we prove the asymptotic behavior of these functions for a periodic-finite-type shift.
Azmeer Nordin, Mohd Salmi Md Noorani
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Prime counting with Composite numbers

, 2023
This paper shows a Möbius expression for the prime counting function as a difference between a natural number n and the composite numbers less than or equal to the number n and transforms this expression towards Riemann’s prime counting function.
Geert Vermeir (6787121)
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Explicit formulas for the Euler’s phi function and the prime counting function

, 2021
An explicit formula which characterizes the pairs of integers that are relatively prime (that is, without common prime factors) is obtained here. It doesn’t require the knowledge of the prime factors of the arguments.
Carlos Mañas Bastidas, Carlos Mañas
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Partial Key Attack Given MSBs of CRT-RSA Private Keys

Mathematics, 2020
The CRT-RSA cryptosystem is the most widely adopted RSA variant in digital applications. It exploits the properties of the Chinese remainder theorem (CRT) to elegantly reduce the size of the private keys.
Amir Hamzah Abd Ghafar +3 more
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