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On the non-existence of a general Benford’s law. (English) Zbl 1140.60003

Benford’s law states that in arbitrary collected ensemble of (real world) data certain digits appear more often as leading digits than others. More formally the mantissae follow a logarithmic distribution accordingly to the base representation that is chosen. In these short communication the author shows by a simple contradiction that if there was probability measure such that fulfills Benford’s law simultaniously for an unbounded number of bases this leads to a contradiction.

MSC:

60A10 Probabilistic measure theory
60E10 Characteristic functions; other transforms
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Full Text: DOI

References:

[1] Benford, F., The law of anomalous numbers, Proceedings of the American Philosophical Society, 78, 551-572 (1938) · Zbl 0018.26502
[2] Hill, T. P., Base-invariance implies Benford’s law, Proceedings of the American Mathematical Society, 123, 887-895 (1995) · Zbl 0813.60002
[3] Hill, T. P., A statistical derivation of the significant-digit law, Statistical Science, 10, 354-363 (1996) · Zbl 0955.60509
[4] Newcomb, S., Note on the frequency of use of the different digits in natural numbers, American Journal of Mathematics, 4, 39-40 (1881) · JFM 13.0161.01
[5] Raimi, R. A., The first digit problem, American Mathematical Monthly, 83, 521-538 (1976) · Zbl 0349.60014
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