Ivan Shishkin, Rye (1878)

Problems/Real analysisUnreviewed

une équation fonctionnelle pour ζ\zeta

by mathman·
30
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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Sachant que pour x>1x>1 :

Γ(x)ζ(x)=∫0+∞tx−1et−1dt\displaystyle\Gamma(x)\zeta(x)=\int_{0}^{+\infty}\frac{t^{x-1}}{e^{t}-1} \mathrm{d}t

Montrez que pour 1<x<21<x<2 :

ζ(x)=1x−1+∑k=1+∞(−1)k−1ζ(x+k)Γ(x+k)Γ(x)Γ(k+2)\displaystyle\zeta(x)=\frac{1}{x-1}+\sum_{k=1}^{+\infty}(-1)^{k-1}\zeta(x+k)\frac{\Gamma(x+k)}{\Gamma(x)\Gamma(k+2)}

Observez que la somme du second membre converge pour tout x<2x<2

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