On the First Occurrence of Values of a Character

Let $\chi$ be a character of order $k (\bmod n)$, and let ${g_m}(\chi )$ be the smallest positive integer at which $\chi$ attains its $(m + 1)$st nonzero value. We consider fixed k and large n and combine elementary group-theoretic considerations with the known results on character sums and sets of integers without large prime factors to obtain estimates for ${g_m}(\chi )$.