values inside this $\beta$-window were used for the optimized combination (\[eq:30\]), the contributions of the other values were suppressed by giving them zero weights.

We also implemented the optimized histogram combination discussed in ref. [@r21] to cross-check the results obtained by (\[eq:30\]) for the specific heat. The Fig. 1(c) shows the weights for the optimal combination of the primary energy histograms according to the prescription of ref. [@r21], i.e., which gives the optimal combination of the three estimates for $N(E) \equiv \rho (E) \Delta E$. Both methods gave comparable results within the statistical errors. We preferred our optimization procedure over the optimized histogram addition because it is simpler to apply to quantities involving constant energy averages such as $\langle\!
\langle m \rangle\!\rangle(E)$, and, more importantly, minimizes the error on each observable of interest separately.

Results at criticality and finite-size scaling analysis
=======================================================

We investigated in our MC study simple cubic lattices of volume $V=L^3$, where $L$=12, 16, 20, 24, 32, 40, and 48. For each $L$ we have made at least three simulations at three different temperatures compiled in Table 1. For all $L$ we took $\beta=0.6929$, because this is the critical inverse temperature found in the recent study of 