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

摘要

A positive integer n is called an r-full integer if for all primes p vertical bar n we have p(r) vertical bar n. Let p be an odd prime. For gcd(n, p) = 1, the smallest positive integer f such that n(f) equivalent to 1 (mod 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