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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.

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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 $a$.

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  • $\begingroup$ Are you a Putnam fellow? $\endgroup$ – Vidyanshu Mishra Jun 7 '17 at 18:03
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    $\begingroup$ Based on nothing, I would not be surprised if a closed form existed for $a = \phi$. $\endgroup$ – marty cohen Jun 7 '17 at 19:33
  • $\begingroup$ @VidyanshuMishra Yes, that was me in 1971. $\endgroup$ – Robert Israel Jun 7 '17 at 20:09
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    $\begingroup$ 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 $a$. $\endgroup$ – Sungjin Kim Jun 7 '17 at 20:38

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