First Formula using the Digamma Function

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

This is one of the most relevant connections between the Riemann Zeta and the Digamma Function, first published by N.G. De Bruijn in 1937. Its demonstration showcases arguments that are useful through all analytic number theory.

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First Formula using the Digamma Function

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

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

Any question?

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