# Sum of the fractional series: $\{1\cdot a\}+\{2\cdot a\}+\{3\cdot a\}+\cdots+\{n\cdot a\}$

Here $\{x\}$ denotes the fractional part of x. In the series $\{1\cdot a\}+\{2\cdot a\}+\{3\cdot a\}+\cdots+\{n\cdot a\}$, $a$ is an irrational number. I need to calculate the sum of this series.

The constant $a$ can be $1.414\ldots$, or $e$ (Euler number), etc. Also, the value of $n$ can be very large, say of the order of $2^{10000}$. So I am guessing there are some properties or theorems which might help to calculate the above but I don't have the required knowledge of them at the moment.

The equidistribution theorem says these fractional parts are equidistributed on $[0,1]$, so in particular your sum is asymptotic to $n/2$. It's possible that more details on the asymptotics might be found. But if you're looking for a "closed form", I doubt that such a thing exists for any irrational $a$.
• Based on nothing, I would not be surprised if a closed form existed for $a = \phi$. – marty cohen Jun 7 '17 at 19:33
• By Erdos-Turan inequality and Koksma inequality, we have $n/2+ O(n^{1-\frac1{\mu-1}+\epsilon})$ where $\mu$ is the irrationality measure of $a$. – Sungjin Kim Jun 7 '17 at 20:38