On an error term related to the Jordan totient function J < inf > k < /inf > (n)

Abstract

We investigate the error terms Ek(x)= ∑ n≤xJk(n)- xk+1 (k+1)ζ(k+1) for k≥2, where Jk(n) = nkΠp|n(1 - 1 pk) for k ≥ 1. For k ≥ 2, we prove ∑ n≤xEk(n)∼ xk+1 2(k+1)ζ(k+1). Also, lim inf n→∞ Ek(x) xk≤ D ζ(k+1), where D = .7159 when k = 2, .6063 when k ≥ 3. On the other hand, even though lim inf n→∞ Ek(x) xk≤- 1 2ζ(k+1), Ek(n) > 0 for integers n sufficiently large. © 1990.