The Hierarchy of Weighted Densities and the First Digit Phenomenon
Résumé
The aim of this paper is to clarify some basic and useful facts about the various notions of densities commonly used in papers on the mantissa distribution of sequences of numbers. We focus on the weighted densities and their hierarchy. Our main tools are classical properties of summation methods. Equivalence classes of weighted densities are identied and their hierarchy is stated. This permits us to give clear answers to several questions about the rst digit phenomenon. Moreover, however light the weights are, we exhibit an example of a sequence of positive numbers whose mantissae do not admit any distribution in the sense of the corresponding density.
Domaines
Mathématiques [math]
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