Representation by Euler-Maclaurin Formula
\zeta(s)=\sum_{n=1}^{N}\frac{1}{n^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-[x]}{x^{s+1}}dx
\Downarrow
\zeta(s)=\frac{1}{s-1}+\frac{1}{2}-s\int_{1}^{\infty}\frac{x-[x]-\frac{1}{2}}{x^{s+1}}dx=\frac{1}{s-1}+\frac{1}{2}-s\int_{1}^{\infty}\frac{\overline{B}_1(x)}{x^{s+1}}dx
An interesting expression of the Riemann Zeta as a finite sum plus an integral, its demonstration relies heavily on Euler’s Summation Formula. It implies a handy connection between the Riemann Zeta and the First Bernoulli Polynomial.
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