Benford or not Benford: new results on digits beyond the first

In this paper, we will see that the proportion of d as p^th digit, where p>1 and d in [[0,9]], in data (obtained thanks to the hereunder developed model) is more likely to follow a law whose probability distribution is determined by a specific upper bound, rather than the generalization of Benford's law to digits beyond the first one. These probability distributions fluctuate around theoretical values of the distribution of the p^th digit of Benford's law. Knowing beforehand the value of the upper bound can be a way to find a better adjusted law than Benford's one.