Riemann辅助函数。基本结果
Riemann's auxiliary Function. Basic Results
AI总结：
本文介绍了Riemann辅助函数$\mathop{\mathcal R }(s)$的定义、性质、积分表达式及在整数处的值，并给出了其与Riemann Zeta函数相关的表达式。
摘要：
我们给出Riemann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$$π^{-s/2}Γ(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-πi s/4}}{ s}\int_{-1}^{-1+i\infty} τ^{s/2}\vartheta_3'(τ)\\,dτ。$$这些结果大多已知，但可作为参考。我们给出$\mathop{\mathcal R }(s)$在整数处的值，但奇自然数处除外。我们有$$ζ(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$其中$\vartheta(t)$、$Z(t)$和$Y(t)$为实函数。
英文摘要：
We give the definition, main properties and integral expressions of the auxiliary function of Riemann $\mathop{\mathcal R }(s)$. For example we prove $$π^{-s/2}Γ(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-πi s/4}}{ s}\int_{-1}^{-1+i\infty} τ^{s/2}\vartheta_3'(τ)\,dτ.$$ 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 $$ζ(\tfrac12+it)=e^{-i\vartheta(t)}Z(t),\quad \mathop{\mathcal R }(\tfrac12+it)=\tfrac12e^{-i\vartheta(t)}(Z(t)+iY(t)),$$ with $\vartheta(t)$, $Z(t)$ and $Y(t)$ real functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
