Abstract
We find estimates for short sums of the form ∑ n m ⩽ X χ 1 ( n ) χ 2 ( m ) \sum _{nm \leqslant X} \chi _1 (n) \chi _2 (m) , where χ 1 \chi _1 and χ 2 \chi _2 are non-principal Dirichlet characters to modulus q q , a cube-free integer, and X X can be taken as small as q 1 2 + ϵ q^{\frac 12+\epsilon } .
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