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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?

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Ooh, I am. Thanks! – Carolus Jul 3 '12 at 10:16

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