In Titchmarsh Theorem 4.11 we find that for $s=\sigma+it$
$$\zeta(s)=\sum_{n\leq x}\frac{1}{n^s}-\frac{x^{1-s}}{1-s}+O\left(\frac1{x^\sigma}\right),$$ when $|t|<\frac{2\pi x}{c}, c>1, \sigma>0$.
I am interested in the approximation of $\zeta$ function when $t=0$. So, am I correct to understand that $$\zeta(\sigma)=\sum_{n\leq x}\frac{1}{n^\sigma}-\frac{x^{1-\sigma}}{1-\sigma}+O\left(\frac1{x^\sigma}\right)?$$