Weird number

Abstract: Let p{ — j tend to infinity with n. We show that log log ^, as /' goes through the indices for which p. is intermediate, forms a limiting Poisson process in the sense of natural density. Let Pi +00). In particular, the simplest results of probabilistic number theory (see Elliott (2, Introduction)) imply that, with density one, co(«)/loglog« is asymptotically one. Hence, with density one, we can distinguish three types of prime divisors: we call pi small if j is bounded as n -» +00, pj large if co — j remains bounded, and all others intermediate. For the investigation of the small prime divisors, tools of elementary number theory suffice. Large prime divisors require special tools, but very old results (due to Dickman, see De Koninck and Ivic (1) for accurate statements and for asymptotic formulas involving large prime divisors) tell us that (log/7-)/logn falls into the interval (a, b), 0 < a < b < 1, with positive density for j = co. Extensions are also known for all large prime divisors, and the results are similar in nature. This perhaps explains why it was 'necessary' and so successful in probabilistic number theory to truncate additive functions at r = r(N) with (logr)/log N -* 0: it simply cancels the effect of the large prime divisors (see Elliott (3), particularly Chapter 12). It indeed required a completely new method of attack when the truncation was abandoned and new types of results were obtained (once again, see (3)). The truncation methods, in which the intermediate prime divisors contributed all the influence for the validity of a statement, already show that the intermediate prime divisors behave asymptotically as independent random variables. The fact that this asymptotic independence is even stronger than what follows from