ON THE LEAST n WITH χ(n) ≠ 1

Suppose that a > 1. By using a method of Linnik employing his Large Sieve one may derive the following result. The number of primitive Dirichlet characters χ with conductor ≤Q such that χ(n) = 1 for all n ≤ (log Q)^a is O(Q^2/a+ε). We improve the exponent 2/a for a > 2 by using a refined version by Heath-Brown of the Halasz–Montgomery ‘Large Values method’.