Denote by $\zeta$ the Riemann zeta function. For $\Re(s)=\sigma>0$, it is well known that

$$\sum_{n\leq x} n^{-s} = \zeta(s) + \frac{x^{1-s}}{1-s}+ O(x^{-\sigma}).$$

But do there exist infinitely many $x$ such that

$$ \Bigg|\sum_{n\leq x} n^{-s} - \zeta(s) - \frac{x^{1-s}}{1-s} \Bigg| \gg x^{-\sigma} ?$$

**ADDENDUM:** Since the left-hand side is equal to $\Big|s\int_{x}^{\infty} \lbrace u \rbrace u^{-s-1}\mathrm{d}u\Big|$, the problem amounts to finding a lower bound for this integral, where $\lbrace y \rbrace$ denotes the fractional part of $y$.