Riemann-Siegel's approximate functional equation

$\zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) $

is the starting point for accurate numerical estimates of $\zeta(s)$ as a function of the partial sums in s and 1-s (of its respective infinite series representations, which inside the critical strip are individually not convergent).

It represents also a best approach for estimating the asymptotic behavior as $t\rightarrow \infty$.

The asymptotic expression for the error term

$E(s) = O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) $

is obtained under the assumptions $0\leq \sigma \leq 1 , \; x,y,t>C>0$, and $2\pi xy=t$.

I am however interested in a somewhat different question:

fixed $t>0$ and $\sigma \neq \frac{1}{2}$, at what rate does the sums of said s and 1-s partial sums approach $\zeta(s)$ ?

In other words, I am investigating the rate of vanishing of

$\sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ - \ \zeta(s) \; = \; O( ? ) \; \; \; \; \; \; x=y \rightarrow \infty $

I am having difficulties in searching through the available literature, as it appears that the assumption $2\pi xy=t$ is usually introduced quite early in the proof of the above asymptotic expression (see for example A. Ivic, The Riemann Zeta-Function, Theorem 4.1), so that I am not quite sure on how to proceed if said assumption is instead left out ...

Could anybody help with suggestions on how to proceed, or about where to look for relevant literature?
Thanks.