One kind of quasi-probability distribution we can impose on the integers (or let's say the positive integers Z+) is as follows: Define Prob(n * Z+) = 1/n for all n in Z+. Then in particular the probability that a "random" integer is squarefree is the Prob(for all p prime, n is not divis. by p^2) = [assuming independence of divisibility by powers of different primes] = Product over p prime of Prob( n not divis. by p^2) = = Product over p prime of (1 - 1/p^2) = = 1 / Product over p prime of ((1 - 1/p^2)^-1) = [by the euler product for Zeta] = 1 / Zeta(2) = 6/pi^2. --Dan Asimov Regarding statements about random integers, I am reminded of certain abuses of formal notation. For instance, in the statement lim 1/x = 0 x->inf we have a statement of equality in which both sides are defined and equal, whereas lim x^2 = inf x->inf is not actually a statement of equality, and neither side is well-defined. Rather, the statement is taken as a whole to assert that x^2 grows without bound as x grows without bound. This is arguably an acceptable abuse of the equality notation. (Possibly we can make this into an actual equality by augmenting R or by some other means, but let's not go there, that is not my point). Similarly, a statement such as "the probability that an integer picked at random is squarefree is 6/pi^2" cannot be taken as equating two well-defined values, but rather the statement must be interpreted as a whole to mean something else, that is, the "the density of squarefree numbers over the integers is 6/pi^2". I understand that this is the common interpretation, though I am rather more disturbed by this abuse of the notion of probability than I am with the abuse of the notion of limit cited above, since the implied uniform distribution over the integers does not exist, and any existing distribution over the integers is likely to invalidate the statement. For example, if an integer is chosen from a normal distribution with mean 0 and variance 1, is its probability of squarefreeness 6/pi^2? Or if we asked human subjects to choose random integers with no other prompting than "pick a number" (another possible interpretation of "integer chosen at random"), would the number chosen be squarefree with probability 6/pi^2? Anyway, I had a thought about a possible definition for random integer. For integer n >= 0, let D[n] be a uniform distribution over the integers on the range [-n, n]. For a set of integers S, let p_S[n] be the probability that a random integer drawn from D[n] is in S. Then the density of S over Z is just the limit n->inf p_S[n]. To interpret the probability that an integer is in S to be the density of S over Z is more or less like saying that D[n] approaches a hypothetical uniform distribution on Z as n increases. Suppose rather, that we let D[n] (n >= 1) be a discrete normal distribution on Z with mean 0 and variance n. We might then also assert that D[n] approaches a hyprothetical uniform distribution on Z as n increases. Does this definition of D[n] lead us to the same probability of an integer being in S as does the prior definition? If so, can we formalize the notion that a sequence of distributions approaches a hypothetical uniform distribution on Z? Can this be used to rigorize the notion of a pseudouniform distribution on Z? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun