Regions without zeros for the auxiliary function of Riemann

Abstract

We give explicit and extended versions of some of Siegel's results. We extend the validity of Siegel's asymptotic development in the second quadrant to most of the third quadrant. We also give precise bounds of the error; this allows us to give an explicit region free of zeros, or with only trivial zeros. The left limit of the zeros on the upper half plane is extended from $1-\sigma\ge a t^{3/7}$ in Siegel to $1-\sigma\ge A t^{2/5}\log t$. Siegel claims that it can be proved that there are no zeros in the region $1-\sigma\ge t^\varepsilon$ for any $\varepsilon>0$. We show that Siegel's proof for the exponent $3/7$ does not extend to prove his claim.