Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and $x_n = p_n^r $, n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.
A new version of Erdos-Turan's inequality is described. The purpose of the present paper is to show that the inequality provides better upper bounds for the discrepancies of some sequences than usual Erdos-Turan's inequality.1991 Mathematics subject classification (Amer. Math. Soc): primary 11K38.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.