Conditional on GRH and negative moment bounds, e^{-y/(2k)}∑_{n≤e^y} μ^(k)(n)χ(n) (or a modified character) has a logarithmic limiting distribution, and the sums obey an almost-all O(x^{1/(2k)}(log x)^{1/2+ε}) bound.
Modified Dirichlet Character Sums over thek-free Integers
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Distribution of sums involving Dirichlet characters over the $k$-free integers
Conditional on GRH and negative moment bounds, e^{-y/(2k)}∑_{n≤e^y} μ^(k)(n)χ(n) (or a modified character) has a logarithmic limiting distribution, and the sums obey an almost-all O(x^{1/(2k)}(log x)^{1/2+ε}) bound.