Riemann's auxiliary Function. Basic Results

Abstract

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

12 pages, 2 figures