# IJPAM: Volume 5, No. 3 (2003)

**RELATIONSHIPS BETWEEN THE INTEGER**

CONDUCTOR AND K-TH ROOT FUNCTIONS

CONDUCTOR AND K-TH ROOT FUNCTIONS

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 -th root is the largest
-th power divisor, and the upper -th root is the smallest -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 -th roots in two
different ways as 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 -th root, Dirichlet series

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2003

**Volume:** 5

**Issue:** 3