Acta Math. Hungar. 68 (4) (1995), 353-361. DISTRIBUTION OF THE VALUES OF q-ADDITIVE FUNCTIONS ON POLYNOMIAL SEQUENCES N. L. BASSILY (Cairo) and I. K.~TAI (Budapest), member of the Academy To Professor K. Tandori on his seventieth birthday 1. Introduction 1.1. As usual N, R, C denote the set of natural, real, complex numbers, respectively, No = N U {0}. ;o is the set of primes, a general element of which is denoted by p. ~r(x) denotes the number of primes up to x. 1.2. Let q N, q > 2 be fixed, E = {0, 1,..., q - 1}. The q-ary expan- sion of n No is defined by (1.1). n = Eaj(n)qJ' aj(n) E j--O The right hand side of (1.1) is clearly a finite sum, since aj(n) = 0 for qJ > n. A function f : No --* R is said to be q-additive if f(0) = 0 and oo (1.2) f(n) = E f(aj(n)qJ). j=O A special q-additive function is a(n) := ~ aj(n), the sum of digit function. Let Aq be the set of q-additive functions. 1.3. The letters N, L are preserved for denoting N = [1~ L = log x. [logq] ' We shall write furthermore e(y) instead of e2~iy. 1.4. Let P(x) be an arbitrary polynomial with integer coefficients, the leading term of which is positive. Let r = deg P(x). 1.5. Let 1 mk := - ~ f(bqk), q beE 1 E f2(bqk) -- m~, bEE 0236-5294/95/$4.00 (~) 1995 Akad6miai Kiad6, Budapest