Riemann's auxiliary Function. Basic Results

We give the definition, main properties and integral expressions of the auxiliary function of Riemann $(s)$. For example we prove $${\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)(s)=-\frac{e^{-\pi is\mathrm{/}4}}{s}{\int }_{-1}^{-1+i\mathrm{\infty }}{\tau }^{s\mathrm{/}2}{\vartheta }_3^{\mathrm{\prime }}(\tau )d\tau \mathrm{.}$$ Many of these results are known, but they serve as a reference. We give the values of $(s)$ at integers except at odd natural numbers. We have $$\zeta (\frac{1}{2}+it)=e^{-i\vartheta (t)}Z(t),(\frac{1}{2}+it)=\frac{1}{2}{e}^{-i\vartheta (t)}(Z(t)+iY(t)),$$ with $\vartheta (t)$, $Z(t)$ and $Y(t)$ real functions.