4.2.4 DISTRIBUTION OF FLOATING POINT NUMBERS 247 is (Database web hosting)
4.2.4 DISTRIBUTION OF FLOATING POINT NUMBERS 247 is the coefficient of .P+ in the function ~C(z)lOZ 10 (24) +rlO( J(&)-ST This condition holds for all values of m, so (24) must equal C(Z), and we obtain the explicit formula –z (lo/r)*–1 -1 C(z) = G 102-l -1 . (25) ( > We want to study asymptotic properties of the coefficients of C(Z), to complete our analysis. The large parenthesized factor in (25) approaches ln(lO/r)/ In 10 = 1 - log,, r as z + 1, so we see that 1 - log,, r C(z) + 1 _ z = R(z) is an analytic function of the complex variable z in the circle I.4 < 11+ g/- In particular, R(z) converges for z = 1, so its coefficients approach zero. This proves that the coefficients of C(z) behave like those of (log,, T - l)/(l -z), that is, lim c, = log,, r -1. WI-CX Finally, we may combine this with (22), to show that Qm(s) approaches 1+ logloi - (1 + Ins + $(ln 3) + . . .) = l%o f uniformly for 1 5 s 2 10. 1 Therefore we have established the logarithmic law for integers by direct calculation, at the same time seeing that it is an extremely good approximation to the average behavior although it is never precisely achieved. The above proofs of Lemma Q and Theorem F are slight simplifications and amplifications of methods due to B. J. Flehinger, AMM 73 (1966), 1056-1061. Many authors have written about the distribution of initial digits, showing that the logarithmic law is a good approximation for many underlying distributions; see the survey by Ralph A. Raimi, -83 (1976), 521-538, for a comprehensive review of the literature. Another interesting (and different) treatment of floating point distribution has been given by Alan G. Konheim, Math. Comp. 19 (1965), 143-144. Exercise 17 discusses an approach to the definition of probability under which the logarithmic law holds exactly, over the integers. Furthermore, ex- ercise 18 demonstrates that any reasonable definition of probability over the integers must lead to the logarithmic law, if it assigns a value to the probability of leading digits.