Sum of fractional parts of linear and logarithmic functions

Question

Fix real numbers $\alpha, \beta \geq 0$ with $\alpha$ irrational, and let $N$ be a positive integer. For $x \in \mathbb{R}$, let $\{x\} := x - \lfloor x \rfloor$ denote the fractional part of $x$. Are there any estimates known for the following sums \begin{equation} S_1 := \sum_{n=1}^N \left(\{\alpha n + \beta\} - \dfrac{1}{2}\right) \end{equation} and \begin{equation} S_2 := \sum_{n=1}^N \left(\{\log(\alpha n + \beta)\} - \dfrac{1}{2}\right). \end{equation} Is it true that $S_1 = o(N)$ and $S_2 = o(N)$ as well? What are methods that may be useful for estimating such sums? Thanks a lot.