On the first occurrence of values of a character

Abstract

Let χ \chi be a character of order k ( mod n ) k\,(\bmod \, n) , and let g m ( χ ) {g_m}(\chi ) be the smallest positive integer at which χ \chi attains its ( m + 1 ) (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 ( χ ) {g_m}(\chi ) .