Riemann's auxiliary Function. Basic Results

We give the definition, main properties and integral expressions of the auxiliary function of Riemann R (s). For example we prove π-s/2Γ(s/2) R (s)=-e-π i s/4 s∫₋₁-1+i∞ τs/2ϑ₃'(τ)\,dτ. Many of these results are known, but they serve as a reference. We give the values of R (s) at integers except at odd natural numbers. We have ζ(12+it)=e-iϑ(t)Z(t), R (12+it)=12e-iϑ(t)(Z(t)+iY(t)), with ϑ(t), $Z(t)$ and $Y(t)$ real functions.