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mathematical analysis > real analysis > real-valued function > piecewise function > Chebyshev function

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Chebyshev function

In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function $ϑ (x)$ or $θ (x)$ is given by
${\\displaystyle \\vartheta (x)=\\sum _{p\\leq x}\\log p}$
where ${\\displaystyle \\log }$ denotes the natural logarithm, with the sum extending over all prime numbers $p$ that are less than or equal to $x$ . The second Chebyshev function $ψ (x)$ is defined similarly, with the sum extending over all prime powers not exceeding $x$
${\\displaystyle \\psi (x)=\\sum _{k\\in \\mathbb {N} }\\sum _{p^{k}\\leq x}\\log p=\\sum _{n\\leq x}\\Lambda (n)=\\sum _{p\\leq x}\\left\\lfloor \\log _{p}x\ ight\ floor \\log p,}$
where $Λ$ is the von Mangoldt function. The Chebyshev functions, especially the second one $ψ (x)$ , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, $π (x)$ (see the exact formula below.) Both Chebyshev functions are asymptotic to $x$ , a statement equivalent to the prime number theorem.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Chebyshev_function)

piecewise function

fonction de Tchebychev

French

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http://data.loterre.fr/ark:/67375/PSR-DT18GRDQ-T

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