Pascal Sebah

ROOTS OF UNITY AND NULLITY MODULO n

For a fixed positive integer l, we consider the function of n that counts the number of elements of order l in ℤ * n . We show that the average growth rate of this function is C l (log n) d(l)―1 for an explicitly given constant C l , where d(l) is the number of divisors of l. From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo n of order l is (d(l) ― 1)C l (log n) d(l)―2 for l ≥ 2. We also consider the number of elements of ℤ n whose lth power equals 0, showing that its average growth rate is D l (log n) l―1 for another explicit constant D l . Two techniques for evaluating sums of multiplicative functions, the Wirsing—Odoni and Selberg— Delange methods, are illustrated by the proofs of these results.