Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the ring Rd of integers of K is not a unique-factorization domain. For d 〈 0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = -1,-2,-3,-7,-11,-19,-43,-67,-163. Let Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.

An Additive Function on a Ring of Integers in the Imaginary Quadratic Field Q(d^(1/2))with Class-Number One

Let Bα（α）be an additive function on a ring of integers in the quadratic number field Q（（1/2）d）given by Bα（α）=∑p丨α*Nα（p）with a fixed α】0,where the asterisk means that the summation is over the non-associate prime divisors p of an integer α in Q（（1/2）d）,N（α）is the norm of α.In this paper we obtain the asymptotic formula of ∑N（α）≤x *Bα（α）in the case where the class-number of Q（（1/2）d）is one.