Required order of the Riemann–Siegel error expansion to certify the nth zero

Determine, for each index n corresponding to a zero of the Hardy Z-function Z(t), the order k(n) of the asymptotic expansion of the Riemann–Siegel error term R(t) (as in equation (RS-app1)) that must be evaluated to confirm the validity of the Riemann Hypothesis at the nth zero of Z(t).

Background

The paper explains that while the Riemann–Siegel formula enables efficient computation of Z(t), analytically tracking zero locations is hindered by the complex structure of its error term, involving coefficients without closed forms.

For some zeros, the Hardy–Littlewood approximate functional equation alone is insufficient, necessitating precise evaluation of the Riemann–Siegel error. Determining the necessary order of this evaluation per zero is highlighted as an unresolved issue.