Log zeta approximation and staircase function

Clear[x, t];
Show[Table[
  Plot[Im[Zeta[1/2 + I t]^(1/x)]/(1/x)/Pi +
    RiemannSiegelTheta[t]/Pi, {t, 0, 60}], {x, 1, 20}]]
Clear[s];
Integrate[Zeta'[s]/Zeta[s], s]
Clear[t, x];
Series[(Zeta[1/2 + I t])^x, {x, 0, 12}]
Series[(Zeta[1/2 + I t])^(1/x), {x, Infinity, 12}]