Average discrepancy for periodic integrands

In the numerical integration of periodic integrands over the $s$-dimensional unit cube, various performance criteria such as $P_{\alpha}$ and $R$ have previously been used. In this paper, we use a criterion called $L_2$ discrepancy. An analogue of this quantity has previously been used to study the error in the case of non-periodic integrands. For this quantity we obtain expressions for the average in the case of number-theoretic and $2^s$ copy rules. The values of these averages are then compared for roughly the same number of points