[section] 1. Introduction

Let a(n) denote the number of non-isomorphic abelian groups with n elements. This is a well-known multiplicative function such that for any prime p we have a([p.sup.[alpha]]) = P([alpha]), where P([alpha]) is the unrestricted partition function. Let l [greater than or equal to] 1 be a fixed integer. The asymptotic behavior of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1 was first investigated by Ivic[3] who obtained the result

[A.sub.l](x) = [C.sub.l,[alpha]]x + O([x.sup.l/2] log x), (1)

which was improved to

[A.sub.l](x) = [C.sub.l,[alpha]]x + O([x.sup.l/2] [log.sup.-1] x log log x), (2)

by Kratzel[4], where [C.sub.l.[alpha]] is a constant. Note that when l = 1, [A.sub.1] (x) is just the counting function of the square-free numbers.

The well-known Dirichlet divisor problem is to study the asymptotic behavior of the sum [summation over (n [less than or equal to] x)] d(n), where d(n) is the Dirichlet divisor function. Dirichlet first proved

[summation over (n [less than or equal to] x)] d(n) = x (log x + 2[gamma] - 1) + O ([x.sup.1/2]). (3)

The error term O([x.sup.1/2]) was improved by many authors. The latest result reads

[summation over (n [less than or equal to] x)] d(n) = x (log x + 2[gamma] - 1) + O ([x.sup.131/416] [log.sup.29647/8320] x), (4)

which was proved by Huxley [1].

In this paper we shall study the asymptotic behavior of the hybrid mean value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We shall prove the following

Theorem. Let l [greater than or equal to] 1 be any fixed integer. We then have the asymptotic formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [A.sub.1,l], [A.sub.2,l] are computable constants.

[section] 2. Some lemmas

Lemma 2.1. [2] (Eular product) If f (n) is a multiplicative function of n, and the Dirichlet series [[infinity].summation over (n = 1)] f (n) [n.sup.-s] is absolutely convergent on some half plane Rs > [[sigma].sub.0] for some real [[sigma].sub.0], then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Lemma 2.2. Suppose A is an infinite subset of N, c(m) satisfies c(m) [much less than] [m.sup.[epsilon] and F(t) [much less than] [t.sup.1/2], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the infinite series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Proof. Suppose T [greater than or equal to] 2 is any real number. Then

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