# Sum of fractional parts of linear and logarithmic functions

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 $$S_1 := \sum_{n=1}^N \left(\{\alpha n + \beta\} - \dfrac{1}{2}\right)$$ and $$S_2 := \sum_{n=1}^N \left(\{\log(\alpha n + \beta)\} - \dfrac{1}{2}\right).$$ 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.

• I would expect $S_1=\frac{N}{2}+o(N)$, though. It may be the case that $S_2=\frac{N}{2}+o(N)$ as well. – Batominovski Jan 22 '17 at 2:09
• Can your guess be proven? – user152169 Jan 22 '17 at 2:20
• Well, a heuristic argument is that $\big\{\alpha n +\beta\big\}$ and $\big\{\log(\alpha n+\beta)\big\}$ are (probably) uniformly distributed on $[0,1)$. If this is true, then we may even have the estimate $\frac{N}{2}+\mathcal{O}\big(\sqrt{N}\big)$ for both $S_1$ and $S_2$. – Batominovski Jan 22 '17 at 2:24
• Makes a lot of sense. But not true until proven :) – user152169 Jan 22 '17 at 2:30
• After a trial with $\alpha=\sqrt{2}$ and $\beta=0$, $\big\{\log(\alpha n+\beta)\big\}$ does not seem uniformly random. But it may still hold that, now with the new definitiosn, $S_2=o(N)$ (although I wouldn't expect $S_2=\mathcal{O}\big(\sqrt{N}\big)$ anymore). I'm too tired to think about $S_1$ now, but it should not be difficult to show $S_1=\mathcal{O}\big(\sqrt{N}\big)$. – Batominovski Jan 22 '17 at 2:48