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On the generalization of the distribution of the significant digits under computation

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  • In this paper we use the set of all positive integers as a sample space whose probability density function is unknown. Then a generalization of the probability distribution of the most significant digits of the set of all physical constants is obtained on the strength of (i) a very general assumption imposed on the density function of the sample space, and (ii) a generalized invariance principle. The assumption is quite weak in the sense that it merely states that the occurrence of an event containing infinitely many elementary events is not impossible. The invariance principle, as is shown, is equivalent to another principle and to two functional equations. A function is constructed and, on the basis of the two foregoing stipulations that characterize the generalization, it is shown that this function is a unique solution, within a multiplicative positive constant, to another functional equation. The function so constructed serves as a stepping stone in reaching our goal. Having the generalization at our disposal, we deduce from it some of the consequences that are of interest. As it turns out, the deduction gives, on one hand, a proof to two empirical formulas published previously and, on the other, a fairly good agreement with the probabilities of three continuous density functions established in the literatures concerning the distribution of the leading digits under algebraic computation. In concluding the paper, a justification is made as to why a special case of the consequences of our result coincides with the probability of one of the three continuous density functions, even though our function is discrete.
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