IJPAM: Volume 5, No. 3 (2003)

RELATIONSHIPS BETWEEN THE INTEGER
CONDUCTOR AND K-TH ROOT FUNCTIONS

Kevin A. Broughan
Department of Mathematics
School of Computing and Mathematical Sciences
University of Waikato
Private Bag 3105, Hamilton 2001, NEW ZEALAND
e-mail: kab@waikato.ac.nz


Abstract.The conductor of a rational integer is the product of the primes which divide it. The lower $k$-th root is the largest $k$-th power divisor, and the upper $k$-th root is the smallest $k$-th power multiple. This paper examines the relationships between these arithmetic functions and their Dirichlet series. It is shown that the conductor is the limit of the upper $k$-th roots in two different ways as $k$ tends to infinity. The asymptotic order of the partial sums is derived and shown to be linear for the lower and quadratic for each of the upper roots, i.e. the same as for the conductor.

Received: January 15, 2003

AMS Subject Classification: 11A05, 11A25, 11M06, 11N37, 11N56

Key Words and Phrases: integer square root, integer $k$-th root, Dirichlet series

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 5
Issue: 3