ASYMPTOTIC ESTIMATES ON FINITE ABELIAN GROUPS

Abstract. By using Ivić’s methods for general divisor problem and count-ing function of abelian finite groups, we obtain results related to several arithmetic functions. 1. Finite abelian groups and semisimple rings Certain common algebraic structures have enumerating functions whose Dirich-let series have a simple product representation involving the Riemann zeta-function. This fact establishes an analytic approach towards the study of these functions, and provides an application of zeta-function theory. It is a well-known fact that every finite Abelian group can be represented as a direct product of cyclic groups of prime power order, moreover, the representation is unique except for possible rearrangements of the factors. Let the arithmetical function a(n) denote the number on nonisomorphic Abel-ian groups with n elements. It is well known that a(n) is a positive, integer-valued multiplicative function, with the property that a(pk) = P (k) for every prime p and every integer k> 1 (here and later p, p1, p2,... denote primes), where P (k) is the number of unrestricted partitions of k (see [29, pp. 7 and 204]). Thus a(pk) does not depend on p but only on k, so that a(n) is a “prime independent ” function, and moreover a(p) = 1 for every prime p. W. Schwarz and E. Wirsing [35] showed that log a(n) 6 log 5 · pi(A) +O((log n)θ), θ = (log 121) / log 125 < 0.994 with A ∼ (1/4) log n. They also show that there are infinitely many integers n for which log a(n) = log 5 ·pi(A). These results sharpen a result of E. Kratzel [18], who showed that lim supn→∞{log a(n) · (log log n / logn)} = (1/4) log 5