On the mantissa distribution of powers of natural and prime numbers

Abstract

Given a fixed integer exponent r≧1, the mantissa sequences of (n ^r) _n and of ${(p_{n}^{r})}_{n}$, where p _n denotes the nth prime number, are known not to admit any distribution with respect to the natural density. In this paper however, we show that, when r goes to infinity, these mantissa sequences tend to be distributed following Benford’s law in an appropriate sense, and we provide convergence speed estimates. In contrast, with respect to the log-density and the loglog-density, it is known that the mantissa sequences of (n ^r) _n and of ${(p_{n}^{r})}_{n}$ are distributed following Benford’s law. Here again, we provide previously unavailable convergence speed estimates for these phenomena. Our main tool is the Erdős–Turán inequality.