Periodica Mathematica Hungarica Vol. 1 (3), (1971), pp. 209--212, NON COMPLETE SUMS OF MULT]PLICATIVE FUNCTIONS by P. ERDOS and I, KATAI (Budapest) 1. It is well known that 2~#(d)= 0 for alI n~ 1. We are interested din concerning the upper estimate of M(n) = m~x M(n,z) = max ] X #(d) I . z z din d~z Previously it was proved that where ~o(n) denotes the number of different prime factors of n (See [1], [2]). One of us asked in a recent paper [3] whether M(n) has a better upper estimate for almost all n. Explicitly it was asked whether (1.2) Min) ~ 2~'(n) holds for almost all integers n with a suitable constant ~ ~ 1. Now we prove a more general theorem, whence (1.2) ~vill immediately follow. 2. T}{~oR~. Let f(n) be a multiplicative function satisfying the con- ditions.: a) }f(n) I ~ 1; b)Let g denote the set of primes 19forw]~ich f(p) -- -1-, 1 let -~ ..... ~. v~ P Then max I X f(d)]~ 2 ~(n) l <z<n o In for almost all n, where o: is an arbitrary constant ~ 1/' 2. To prove this we need two lemmas. Let x 1 = log x, x2 =-10g xl, Yl ---- log y, Y2 ---- log Yl, .Q(n) be the number of all prime divisors of n counted each of' them by their multiplicity. Let s be an arbitrarily small positive constant, R -- (1 + e)x2. The symbol 2:' denotes a sum extended over those n for which ~Q(n)~R. Since, by the