The sequences of mantissa of positive integers and of prime numbers are known not to be distributed as Benford's law in the sense of the natural density. We show that we can correct this defect by selecting the integers or the primes by means of an adequate random process and we investigate the rate of convergence. Our main tools are uniform bounds for deterministic and random trigonometric polynomials. We then adapt the random process to prove the same result for logarithms and iterated logarithms of integers. Finally we show that, in many cases, the mantissa law of the $n$th randomly selected term converges weakly to the Benford's law.
"Random number sequences and the first digit phenomenon." Electron. J. Probab. 17 1 - 17, 2012. https://doi.org/10.1214/EJP.v17-1900