Representations using Bernoulli Polynomials
\zeta(s)=\sum_{n=1}^{N}\frac{1}{n^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}N^{-s}+\sum_{j=1}^{k}\binom{s+2j-2}{2j-1}\frac{B_{2j}}{2j}N^{1-s-2j}-\binom{s+k}{k+1}\int_{N}^{\infty}\frac{\overline{B}_{k+1}(x)}{x^{s+k+1}}dx
\Downarrow
\zeta(s)=\frac{1}{s-1}+\frac{1}{2}+\sum_{j=1}^{k}\binom{s+2j-2}{2j-1}\frac{B_{2j}}{2j}-\binom{s+k}{k+1}\int_{1}^{\infty}\frac{\overline{B}_{k+1}(x)}{x^{s+k+1}}dx
A clever way to express the Riemann Zeta function with Bernoulli Polynomials, this formula gives a glimpse into the connections of this two seemingly unrelated functions.
Essential to the understanding of the demonstration is the Representation by Euler-MacLaurin Formula.
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.
Didn’t find what you were looking for? Keep looking through all our formulas regarding the Riemann Zeta Function!
