I have been trying to calculate the following limit of a fractional part without much success, as I am not very competent at mathematics. Could anyone provide any help please?
$lim_{x \to \infty} \frac{1}{x+1}\sum_{k=0}^x \{k/c\}$
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What if $c$ is an integer ? What if $c = p/q$ is rational ? – reuns Oct 21 '17 at 01:30
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If c is irrational (but not zero), k/c has uniform distribution in (0,1). Thus the average goes to 0.5. The rational case can be more difficult.
Dave
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The rational case is easy ($p$ is inversible modulo $q$ so $\sum_{k=1}^{mq} { kp/q} =\sum_{k=1}^{mq} { k/q}$). Then for $c$ irrational find a sequence $p_n/q_n$ of rationals such that $\lim_{n \to \infty} q_n |c-\frac{p_n}{q_n}| = 0$ to obtain the result. – reuns Oct 21 '17 at 01:55