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Poussin proof — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Poussin_proof
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T17:23:11.330Z
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Poussin proof — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Poussin_proof (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_dedecd34f640 (Metraipolis archive dc_dedecd34f640, sha256 7828d9e6540be34e87fa3cae45e74340e3bf1bdf725e2756ee041c24d831cfb0)

BibTeX

@misc{dcdedecd34f640,
  title = {Poussin proof — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Poussin_proof},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_dedecd34f640; sha256 7828d9e6540be34e87fa3cae45e74340e3bf1bdf725e2756ee041c24d831cfb0},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Poussin_proof. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In number theory, a branch of mathematics, the Poussin proof is the proof of an identity related to the fractional part of a ratio. In 1838, Peter Gustav Lejeune Dirichlet proved an approximate formula for the average number of divisors of all the numbers from 1 to n: ∑ k = 1 n d ( k ) n ≈ ln n + 2 γ − 1 , {\displaystyle {\frac {\sum _{k=1}^{n}d(k)}{n}}\approx \ln n+2\gamma -1,} where d represents the divisor function, and γ represents the Euler-Mascheroni constant. In 1898, Charles Jean de la Vallée-Poussin proved that if a large number n is divided by all the primes up to n, then the average fraction by which the quotient falls short of the next whole number is γ: ∑ p ≤ n { n p } π ( n ) ≈ 1 − γ , {\displaystyle {\frac {\sum _{p\leq n}\left\{{\frac {n}{p}}\right\}}{\pi (n)}}\approx 1-\gamma ,} where {x} represents the fractional part of x, and π represents the prime-counting function. For example, if we divide 29 by 2, we get 14.5, which falls short of 15 by 0.5.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Number theory»).
Fonte: Wikipedia in inglese, voce «Poussin proof» — https://en.wikipedia.org/wiki/Poussin_proof
Wikidata Q7235754 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q7235754
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Poussin_proof?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1224220626, consultata il 2026-10-07T16:54:19.898Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: a2ca5069ae50ab55842701d9dcc97f6d5648f45b8117c9cbf56327335c722f0b