Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence
TL;DR: In this paper , it was shown that the asymptotic limit of a summation operation performed on a unique subsequence of the prime numbers yields the prime number counting function π ( x ) as x approaches ∞.
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Abstract: A BSTRACT . In mathematics, the prime counting function π ( x ) is defined as the function yielding the number of primes less than or equal to a given number x . In this paper, we prove that the asymptotic limit of a summation operation performed on a unique subsequence of the prime numbers yields the prime number counting function π ( x ) as x approaches ∞ . We also show that the prime number count π ( n ) can be estimated with a notable degree of accuracy by performing the summation operation on the subsquence up to a limit n .
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Citations
An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function
T. Ganesan
TL;DR: This study introduces a novel approximation function ϕ(n) for prime counting, offering a fresh perspective on number-theoretic studies, and reveals modified representations of the Riemann zeta function, Chebyshev function, and prime number theorem with improved accuracy at large magnitudes.
References
•Book
Introduction to analytic number theory
Tom M. Apostol
- 01 Jan 1976
TL;DR: The Mathematics 160 course at the California Institute of Technology as discussed by the authors was the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the University of California during the last 25 years.
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Properties of Higher-Order Prime Number Sequences
TL;DR: In this article, the authors analyzed properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes and introduced a new sieve which will generate these prime numbers via the systematic selection and elimination of prime index on the real number line.
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Properties of Higher-Order Prime Number Sequences
TL;DR: In this paper, the authors analyzed properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes and introduced a new sieve which will generate these prime number sequence via the systematic selection and elimination of the prime number indices on the real number line.
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