An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function

Timothy Ganesan ()
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Timothy Ganesan: Department of Physics & Astronomy, University of Calgary, Calgary, AB T2N 1N4, Canada

Mathematics, 2024, vol. 12, issue 17, 1-11

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 10 100 is less than 1%.

Keywords: prime-counting approximation; prime number theorem; Riemann zeta function; logarithmic integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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