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ROOTS OF UNITY AND NULLITY MODULO n

STEVEN FINCH, GREG MARTIN and PASCAL SEBAH
Proceedings of the American Mathematical Society
Vol. 138, No. 8 (AUGUST 2010), pp. 2729-2743
Stable URL: http://www.jstor.org/stable/20764230
Page Count: 15
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ROOTS OF UNITY AND NULLITY MODULO n
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Abstract

For a fixed positive integer ℓ, we consider the function of n that counts the number of elements of order ℓ in ${\Bbb Z}_{n}^{\ast}$ . We show that the average growth rate of this function is C(log n)d(ℓ)-1 for an explicitly given constant 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(log n)d(ℓ)-2 for ℓ ≥ 2. We also consider the number of elements of ℤn whose ℓth power equals 0, showing that its average growth rate is D(log n)ℓ-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.

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