ROOTS OF UNITY AND NULLITY MODULO n

For a fixed positive integer , we consider the function of n that counts the number of elements of order in Zn. We show that the average growth rate of this function is C (logn) d( )−1 for an explicitly given constant C , where d( ) is the number of divisors of . From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo n of order is (d( )− 1)C (logn)d( )−2 for ≥ 2. We also consider the number of elements of Zn whose th power equals 0, showing that its average growth rate is D (logn) −1 for another explicit constant D . Two techniques for evaluating sums of multiplicative functions, the Wirsing–Odoni and Selberg– Delange methods, are illustrated by the proofs of these results. Let Zn denote the group (under multiplication modulo n) of integers relatively prime to n, and let denote a fixed positive integer. Define a (n) to be the number of solutions of x = 1 in Zn. The value of a (p), when p is prime, ranges over all divisors of ; however, a (n) can be much larger than if n is composite. It is therefore interesting to ask how the function a (n) behaves on average over n. We can ask the same about ã (n), which we let denote the number of solutions of x = 1 in Zn for which x m = 1 for all 1 ≤ m < ; such an x is said to be of order in the group Zn. In other words, a counts the th roots of unity modulo n, while ã counts the primitive th roots of unity modulo n. The following theorem, which is proved in Section 2, gives the average rate of growth for both functions a and ã for every positive integer . In the statement of the theorem, we employ the usual notation p ‖ n to mean that p | n but p n; we also use d( ) to denote the number of divisors of . Theorem 1. For any positive integer and for any real number ε > 0,