The classical Polya-Vinogradov inequality gives a bound (roughly of size
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 Dirichlet characters
which are nothing but characters
of the multiplicative group GL(1, Fp)
of invertible elements in Fp, the finite field of
p elements, we can work with
representations of the group
GL(n, Fp) for n > 1 and try to
generalise the result. I shall describe my joint work with
C. S. Rajan on this question and our result for the case n = 2.
As an application, we will describe a matrix analogue
of the problem of estimating the least primitive root modulo a prime.