Behaviour of $\sum_{k=1}^n\left(\left(\frac{3}{2}\right)^k\ (\mathrm{mod}\ 1)\right)$

Question

Using Mathematica I found that the relation $$\sum_{k=1}^n\left(\left(\frac{3}{2}\right)^k\ (\mathrm{mod}\ 1)\right)\approx\frac{n}{2}$$ seems to hold. Actually, every fraction of the form $\frac{b}{a}$, with $b>a$ and $\mathrm{gcd}(a,b)=1$, seems to behave similarly. Example, $\frac{3}{2}$:

So, can we prove some asymptotic formula or somehow show that this behavior is constant?

Answer 1

You might be interested in this: http://mathworld.wolfram.com/PowerFractionalParts.html