One way would be to calculate it: it is not difficult using a spreadsheet and recursion, and a short piece of R code is shown below.
$\epsilon$ seems to be zero for $n=1,2,3$, to be positive $\frac{1}{648}$ for $n=4$, negative $- \frac{1}{9270}$ for $n=5$, zero for $n=6$, and then to be positive and increasing up to a maximum of about $0.01223$ for $n=52$ after which it starts to decline.
Rather more seriously $\frac{1}{n}$ soon becomes a very poor approximation to whether the sum is divisible by $n$. For $n \gt 43$ it is more than twice the true value and then gets rapidly worse. There is a better alternative.
Added
A better approximation for large $n$ would be to assume the central limit theorem applies well enough and work out the probability of achieving $3$ times the number of rolls or $4$ times the number of rolls ($1$, $2$, $5$, and $6$ times the number of rolls are too unlikely to be worth worrying about). These are half the number of rolls away from the mean, and the variance is $\frac{35}{12}$ times the number of rolls. So a possible approximation for the required probability is
$\sqrt{ \frac{24}{35 \pi n} } \exp\left( - \frac{3 n}{70} \right).$
It is a good approximation for a wide range of values as shown in the chart drawn by the following R code which calculates the actual values. It is a better approximation than $\frac{1}{n}$ for $n \ge 14$.
maxrolls <- 1000 probmultofrolls <- rep(0,maxrolls) rolls <- 1:maxrolls probs <- 1 # after 0 rolls score is zero with probability 1 for (r in rolls) { probs <- (c(0,probs) + c(0,0,probs) + c(0,0,0,probs) + c(0,0,0,0,probs) + c(0,0,0,0,0,probs) + c(0,0,0,0,0,0,probs) ) / 6 probmultofrolls[r] <- sum( probs[r * (1:6) + 1] ) # +1 because started at 0 } plot( probmultofrolls ~ rolls, log="xy" ) lines( 1/rolls ~ rolls, col="blue" ) lines( exp( -3*rolls/70) * sqrt(24 / (35 * pi * rolls) ) ~ rolls, col="red" )
which produces the following chart with logarithmic scales and $1 \le n \le 1000$: black circles are the actual values; the blue line is $\frac{1}{n}$; and the red line is the approximation based on the central limit theorem.

Even this may not hold for even larger $n$, since the central limit theorem is often not a good approximation in the extreme tails of a distribution. It is within 5% of the true figure when $9 \le n \le 191$.