On the distribution of square-full and cube-full primitive roots

Abstract

A positive integer n is called an r-full integer if for all primes $p\mid n$ we have $p^r\mid n.$ Let p be an odd prime. For $\gcd (n,p)=1$, the smallest positive integer f such that $n^f\equiv 1\pmod p$ is called the exponent of n modulo p. If $f=p-1$ then n is called a primitive root modulo p. Let $T_r(n)$ be the characteristic function of the r-full primitive roots modulo p. In this paper we derive the asymptotic formula for the following sums

$$\begin{aligned} \sum _{n\le x}T_2(n),\quad \sum _{n\le x}T_3(n), \end{aligned}$$

by using properties of character sums.