Acta Mathematica Academiae Scientiarum Hungaricae 67. (1995)
1995 / 1-2. szám - Manstavičius, E.: Functional approach in the divisor distribution problems
Acta Math. Hangar. 67 (1-2) (1995), 1-17. FUNCTIONAL APPROACH IN THE DIVISOR DISTRIBUTION PROBLEMS E. MANSTAVIÖIUS* (Vilnius) 1. Results One-dimensional laws of the iterated logarithm for additive arithmetic functions have been investigated in [1], [9] and in several papers by the author. Our main results were summarized in [13]. Independently, R. R. Hall and G. Tenenbaum [7] considered a partial case and gave a new numbertheoretical application. On the other hand, the functional limit theorems for arithmetical processes have a fairly large literature too. We mention here only [14] and [20] containing the prehistory of the subject. In the present remark based upon the Kubilius probabilistic approach [9] we present the functional law of iterated logarithm of Strassen type (see [10], [19], and [21]). Let us observe one inconvenient feature of the theorems concerning the strong convergence of sequences of arithmetical functions or arithmetical processes. Usually, we have a sequence of probabilistic spaces which is difficult to inbed into some fixed and at the same time not too complicated space. As far as we know, the authors who have dealt with the law of iterated logarithm for sums of independent random variables (i.r.v.’s), except W.Feller [5] did not pay any attention to the origin of the common probability space. In some number theoretical problems, contrary to our case (see, for instance, [17]) the product space is quite appropriate. To get over this obstacle, we [11] have suggested a new approach. Our concept is based on the following definitions. Let (S,d) be a separable metric space and Y, Ух,...,Уп be S-valued random elements all defined on the probability space {fin, Tn, Pn}, n = 1,2,... . Let, as usual, d(X, A) = inf { d(X, Z) : Z E Aj , A C S. * The final version of the paper has been prepared during my visit to the Mathematical Institute of the Hungarian Academy of Sciences. Financially supported by Hungarian National Foundation for Science, Grant No 1901 (Number Theory). 0236-5294/95/$4.00 © 1995 Akadémiai Kiadó, Budapest