Abstract :

The classical Polya-Vinogradov inequality gives a uniform bound (roughly square root of p) on the sum of values of a Dirichlet character modulo p along a segment which is independent of the length of the segment. The proof uses Fourier Analysis on finite abelian groups. Instead of characters of the mutiplicative group GL(1, F_p) of invertible elements in F_p, the finite field of p elements, we can work with representations of the group GL(n, F_p) for n >1 and try to generalise the result. I shall describe my joint work with C.S. Rajan on this question. As an application, we describe a matrix analogue of the problem of estimating the least primitive root modulo a prime.