On the approximation of the zeta function by Dirichlet polynomials

We prove that for $s=\sigma +it$ with $\sigma \ge 0$ and $0<t\le x$, we have $$\zeta (s)=\sum _{n\le x}{n}^{-s}+\frac{x^{1-s}}{(s-1)}+\mathrm{\Theta }\frac{29}{14}{x}^{-\sigma },\frac{29}{14}=2.07142\dots$$ where $\mathrm{\Theta }$ is a complex number with $\mathrm{\mid }\mathrm{\Theta }\mathrm{\mid }\le 1$. This improves Theorem 4.11 of Titchmarsh.