Title: Riemann's auxiliary Function. Basic Results Show Chinese title

Title: 黎曼的辅助函数 基本结果

Abstract: We give the definition, main properties and integral expressions of the auxiliary function of Riemann $\mathop{\mathcal R }(s)$. For example we prove $$\pi^{-s/2}\Gamma(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-\pi i s/4}}{ s}\int_{-1}^{-1+i\infty} \tau^{s/2}\vartheta_3'(\tau)\,d\tau.$$ Many of these results are known, but they serve as a reference. We give the values of $\mathop{\mathcal R }(s)$ at integers except at odd natural numbers. We have $$\zeta(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$ with $\vartheta(t)$, $Z(t)$ and $Y(t)$ real functions. Show Chinese abstract

Abstract: 我们给出黎曼辅助函数$\mathop{\mathcal R }(s)$的定义、主要性质和积分表达式。 例如我们证明 $$\pi^{-s/2}\Gamma(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-\pi i s/4}}{ s}\int_{-1}^{-1+i\infty} \tau^{s/2}\vartheta_3'(\tau)\,d\tau.$$许多这些结果是已知的，但它们作为参考。 我们给出 $\mathop{\mathcal R }(s)$在整数上的值，除了奇自然数。 我们有 $$\zeta(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$与$\vartheta(t)$， $Z(t)$和$Y(t)$为实函数。