Asymptotic Expansions of the auxiliary function

Siegel in 1932 published a paper on Riemann's posthumous writings, including a study of the Riemann-Siegel formula. In this paper we explicitly give the asymptotic developments of $\mathop{\mathcal R }(s)$ suggested by Siegel. We extend the range of validity of these asymptotic developments. As a consequence we specify a region in which the function $\mathop{\mathcal R }(s)$ has no zeros. We also give complete proofs of some of Siegel's assertions. We also include a theorem on the asymptotic behaviour of $\mathop{\mathcal R }(\frac12-it)$ for $t \to+\infty$. Although the real part of $e^{-i\vartheta(t)}\mathop{\mathcal R }(\frac12-it)$ is $Z(t)$ the imaginary part grows exponentially, this is why for the study of the zeros of $Z(t)$ it is preferable to consider $\mathop{\mathcal R }(\frac12+it)$ for $t>0$.