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*Estimare indicatorul lui Euler cu o functie de clasa C^1

 
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PostPosted: Thu Jan 24, 2008 7:20 am    Post subject: *Estimare indicatorul lui Euler cu o functie de clasa C^1 Reply with quote

Iata o mica teorema pe care am intalnit-o intr-un articol publicat in Comm. Math. Phys. de Florin Boca, Alexandru Zaharescu si Radu Gologan. Ea suna cam asa:

Fie f:[a,b]\to\mathbb{R} o functie de clasa C^{1}([a,b]). Sa se arate ca

\sum_{a<k\leq b}\frac{\varphi(k)}{k}f(k)=\frac{6}{\pi^{2}}\int_{a}^{b}f(x)dx+O( \log b (||f||_{\infty}+\int_{a}^{b} |f\prime (x) |dx)),

unde \varphi(n) este indicatorul lui Euler.
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