Computing $\pi (x)$ analytically

DOI10.1090/S0025-5718-2014-02884-6zbMath1336.11078arXiv1203.5712OpenAlexW2027537091WikidataQ56059245 ScholiaQ56059245MaRDI QIDQ5179234

David J. Platt

Publication date: 19 March 2015

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1203.5712

zbMATH Keywords

Computation of exact values of the prime counting function

Mathematics Subject Classification ID

Analytic computations (11Y35) Values of arithmetic functions; tables (11Y70)

Related Items (18)

On equalities involving integrals of the logarithm of the Riemann \(\varsigma\)-function with exponential weight which are equivalent to the Riemann hypothesis ⋮ Improvements to Turing's method. II ⋮ Summing \(\mu(n)\): a faster elementary algorithm ⋮ Explicit zero-free regions for the Riemann zeta-function ⋮ Computations of the Mertens function and improved bounds on the Mertens conjecture ⋮ Isolating some non-trivial zeros of zeta ⋮ An improved explicit bound on \(| \zeta(\frac{1}{2} + i t) |\) ⋮ Explicit bounds on the logarithmic derivative and the reciprocal of the Riemann zeta-function ⋮ New bounds for $\psi (x)$ ⋮ Estimates of $\psi ,\theta $ for large values of $x$ without the Riemann hypothesis ⋮ An improved analytic method for calculating \(\pi(x)\) ⋮ Updating the error term in the prime number theorem ⋮ Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function ⋮ Estimating $\pi (x)$ and related functions under partial RH assumptions ⋮ Sharper bounds for the Chebyshev function 𝜃(𝑥) ⋮ A still sharper region where $\pi (x)-{\mathrm {li}}(x)$ is positive ⋮ Every odd number greater than $1$ is the sum of at most five primes ⋮ A practical analytic method for calculating $\pi (x)$

Uses Software

• MPFI

Cites Work

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• On the exact number of primes less than a given limit
• Motivations for an arbitrary precision interval arithmetic and the MPFI library
• Isolating some non-trivial zeros of zeta
• Computing π(x): The Meissel-Lehmer Method
• Computing π(x): An analytic method
• Prime sieves using binary quadratic forms
• Computing 𝜋(𝑥): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method
• A practical analytic method for calculating $\pi (x)$
• Artin's Conjecture, Turing's Method, and the Riemann Hypothesis
• Explicit Bounds for Some Functions of Prime Numbers

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