We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet character sums.
Mean values of some Hecke characters
Bounds for moments of quadratic Dirichlet character sums of prime moduli
Mean values of divisors of forms $n^2+Nm^2$
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