Distributions of the values of some arithmetical functions

Distributions of the values of some arithmetical functions by P. ERDÖS (Budapest) and A. SCHINZEL (Warszawa) § 1. Y. Wang and A. Schinzel proved. by Brun's method, the following theorem ([3]) : For any given sequence of h nova-negative-numbers a,, a 2 ,. . ., a h and > 0, there exist positive constants e = c(a, E) and x o = x0 (a, E) such that the number of positive integers n, <. x satisfying (ra {-i) 92(n-i:-1)-p < (I. ~ i h) is greater than cxjlog h'T'x, whenever x > x,. They also proved the analogous theorem for the function a. Shao Pin Tsung, also using Brun's method, extended this result to all multiplicative positive functions f,(ir) satisfying the following conditions ([4]) : 1. For any positive integer 1 and prime number p lim (f,(pa) ;pa') = 1 (p denotes primes). 1J->C II. There exists an interval Via, b>, a = 0 or b = oo, such that for any integer-11 > 0 the set of numbers where (_V, ll) = 1, is dense (This formulation is not the same but equivalent to the original one .) In this paper we shall show without using Brun's method that if we replace the condition. I by the condition ~~ff,(p)-ps)_ y 2s+1 (bit preserving condition II) then there exist more than. C(a, e)x posi