Improving Riemann prime counting

Michel Planat and Patrick Solé
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 3, Pages 38—44
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Authors and affiliations

Michel Planat
Institut FEMTO-ST, CNRS,
15 B Avenue des Montboucons, F-25044 Besançon, France

Patrick Solé
Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France.
Mathematics Department, King Abdulaziz University,
Jeddah, Saudi Arabia

Abstract

Prime number theorem asserts that (at large x) the prime counting function π(x) is approximately the logarithmic integral li(x). In the intermediate range, Riemann prime counting function deviates from π(x) by the asymptotically vanishing sum Σ_ρRi(x^ρ) depending on the critical zeros ρ of the Riemann zeta function ζ(s). We find a fit π(x) ≈ Ri⁽³⁾{ψ(x)} (with three to four new exact digits compared to li(x)) by making use of the Von Mangoldt explicit formula for the Chebyshev function ψ(x). Another equivalent fit makes use of the Gram formula with the variable ψ(x). Doing so, we evaluate π(x) in the range x = 10ⁱ, with the help of the first 2×10⁶ Riemann zeros ρ. A few remarks related to Riemann hypothesis (RH) are given in this context.

Keywords

• Prime counting
• Chebyshev psi function
• Riemann hypothesis

AMS Classification

• Primary 11N05
• Secondary 11A25, 11N37

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