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 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 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 R (12-it) for t +. Although the real part of e^-i (t) R (12-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 R (12+it) for t>0.