Regions without zeros for the auxiliary function of Riemann

TL;DR

This paper extends classical results on the auxiliary function of the Riemann zeta function, providing explicit zero-free regions and error bounds, clarifying limitations of Siegel's claims about zero distribution.

Contribution

It offers explicit bounds and extended zero-free regions for the auxiliary function of the Riemann zeta function, refining Siegel's asymptotic results and clarifying the limitations of existing proofs.

Findings

Extended zero-free regions to most of the third quadrant

Provided explicit bounds for error terms

Clarified limitations of Siegel's zero-free region claims

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.