Third Formula using the Digamma Function

\zeta(s)=\frac{1}{(s-1)}+\frac{\sin(\pi s)}{\pi(s-1)}\int_{0}^{\infty}x^{1-s}\left(\frac{1}{1+x}-\psi'(1+x)\right)dx

This formula connecting the Digamma Function and the Riemann Zeta is convenient for isolating the pole at s=1. It appeared first in N.G. De Bruijn’s paper “Integralen voor de zeta -functie van Riemann” dated 1937.

This Relation is a corollary of the Second Formula using the Digamma Function.

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Representation by Euler-Maclaurin Formula

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Third Formula using the Digamma Function

\zeta(s)=\frac{1}{(s-1)}+\frac{\sin(\pi s)}{\pi(s-1)}\int_{0}^{\infty}x^{1-s}\left(\frac{1}{1+x}-\psi'(1+x)\right)dx

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.

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.