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Open AccessArticle
An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function
by
Timothy Ganesan
Timothy Ganesan
Department of Physics & Astronomy, University of Calgary, Calgary, AB T2N 1N4, Canada
Mathematics 2024, 12(17), 2624; https://doi.org/10.3390/math12172624 (registering DOI)
Submission received: 6 July 2024
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Revised: 19 August 2024
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Accepted: 23 August 2024
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Published: 24 August 2024
Abstract
Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%.
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MDPI and ACS Style
Ganesan, T.
An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function. Mathematics 2024, 12, 2624.
https://doi.org/10.3390/math12172624
AMA Style
Ganesan T.
An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function. Mathematics. 2024; 12(17):2624.
https://doi.org/10.3390/math12172624
Chicago/Turabian Style
Ganesan, Timothy.
2024. "An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function" Mathematics 12, no. 17: 2624.
https://doi.org/10.3390/math12172624
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