Large sums of high order characters II

Let χ be a primitive Dirichlet character modulo q , and let δ > 0 . Assuming that χ has large order d , for any d th root of unity α we obtain non-trivial upper bounds for the number of n ≤ x such that χ ( n ) = α , provided x > q δ . This improves upon a previous result of the first author by removing restrictions on q and d . As a corollary, we deduce that if the largest prime factor of d satisfies P + ( d ) → ∞ then the level set χ ( n ) = α has o ( x ) such solutions whenever x > q δ , for any fixed δ > 0 . Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters χ ℓ , averaged over 1 ≤ ℓ ≤ d - 1 , due to the first author, which goes beyond Burgess’ theorem as soon as d is sufficiently large. We in fact show the alternative result that the partial sum of either (a) χ itself, or (b) χ ℓ , for “almost all” 1 ≤ ℓ ≤ d - 1 , exhibits cancellation on the interval 1 , q δ , for any fixed δ > 0 . By an analogous method, we also show that the Pólya–Vinogradov inequality may be improved for either χ itself or for almost all χ ℓ , with 1 ≤ ℓ ≤ <mml:mi