Ivan Shishkin, Rye (1878)

Problems/Number theoryUnreviewed

La formule de Ramanujan-Wilson

by NawzadHogan·
85
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Soient a,b∈Ca,b \in\mathbb{C}.
Montrer que, pour tout s∈Cs\in\mathbb{C} tel que Re⁡(s)>max⁡(1,Re⁡(a)+1,Re⁡(b)+1,Re⁡(a+b)+1)\Re(s)>\max(1,\Re(a)+1,\Re(b)+1,\Re(a+b)+1),

ζ(s)ζ(s−a)ζ(s−b)ζ(s−a−b)ζ(2s−a−b)=∑n=1+∞σa(n)σb(n)ns,\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}=\sum^{+\infty}_{n=1}\frac{\sigma_{a}(n)\sigma_b(n)}{n^s},où ζ\zeta est la fonction zêta de Riemann, et σa(n)=∑d ∣ nda\sigma_{a}(n)=\sum_{d \ | \ n}d^{a}.

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