Tuning up the Kolmogorov–Smirnov test for testing Benford’s law

For moderately large and large numbers of data points (n≥100), the Kolmogorov-Smirnov test is too conservative for testing Benford's law. Moreover, the asymptotic cumulative distribution function of the Kolmogorov statistic shows unacceptable large deviations, up to about 35%, from the ones obtained in Monte Carlo simulations. Such deviations can be reduced to a level below 0.5% if an appropriate linear transformation of the argument of the Kolmogorov cumulative function is performed.