Contour Integrals

\zeta(s)=\frac{\Gamma(1-s)}{2\pi i}\int_{-\infty}^{(0,+)}\frac{t^{s-1}}{e^{-t}-1}dt

\Downarrow

\zeta(s)=\frac{\Gamma(1-s)}{2\pi i(1-2^{1-s})}\int_{-\infty}^{(0,+)}\frac{t^{s-1}}{e^{-t}+1}dt

There are many examples of Formulas for the Riemann Zeta that use Contour Integration, and many of them also relate to the Gamma Function in some way.

These are not trivial and require some effort to prove, but they display a type of reasoning useful in all analytic number theory.

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Representation using Bernoulli Polynomials

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Contour Integrals

\zeta(s)=\frac{\Gamma(1-s)}{2\pi i}\int_{-\infty}^{(0,+)}\frac{t^{s-1}}{e^{-t}-1}dt \Downarrow \zeta(s)=\frac{\Gamma(1-s)}{2\pi i(1-2^{1-s})}\int_{-\infty}^{(0,+)}\frac{t^{s-1}}{e^{-t}+1}dt

Any questions? Feel free to contact our authors at: [email protected], you will be put directly in touch with the person who wrote the text.

\zeta(s)=\frac{\Gamma(1-s)}{2\pi i}\int_{-\infty}^{(0,+)}\frac{t^{s-1}}{e^{-t}-1}dt

\Downarrow

\zeta(s)=\frac{\Gamma(1-s)}{2\pi i(1-2^{1-s})}\int_{-\infty}^{(0,+)}\frac{t^{s-1}}{e^{-t}+1}dt

Any questions?

Feel free to contact our authors at: [email protected], you will be put directly in touch with the person who wrote the text.