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New Fast Methods To Compute The Number Of Primes Smaller Than A Given Value

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Ukrainian Mathematical Journal Aims and scope

We describe new fast algorithms intended for the evaluation of π(x) inspired by the harmonic and geometric mean integrals that can be used on any pocket calculator. In particular, the formula h(x) based on the harmonic mean is within ≈ 15 of the actual value for 3 ≤ x ≤ 10000. The approximation verifies the inequality h(x) ≤ Li(x) and, therefore, is better than Li(x) for small x. We show that h(x) and their extensions are more accurate than the other famous approximations, such as the Locker–Ernst or Legendre approximations also for large x. In addition, we derive another function g(x) based on the geometric mean integral that employs h(x) as an input and allows one to significantly improve the quality of this method. We show that g(x) is within ≈ 25 of the actual value for x ≤ 50, 000 (to compare, Li(x) lies within ≈ 40 for the same range) and asymptotically \(g\left(x\right)\sim \frac{x}{1\mathrm{n} x}\mathrm{exp}\left(\frac{1}{1\mathrm{n} x-1}\right).\)

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Authors and Affiliations

  1. Department of Mathematics, IES Carrús, Elche, Alicante, Spain

    G. R. P. Teruel

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  1. G. R. P. Teruel
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Correspondence to G. R. P. Teruel.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 9, pp. 1264–1273, September, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i9.6193.

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Teruel, G.R.P. New Fast Methods To Compute The Number Of Primes Smaller Than A Given Value. Ukr Math J 74, 1441–1451 (2023). https://doi.org/10.1007/s11253-023-02145-2

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  • DOI: https://doi.org/10.1007/s11253-023-02145-2

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