Character sums over shifted primes and composite modulus

In this talk we describe some new estimates for character sums over shifted primes and composite modulus which improve a series of results due to Friedlander, Gong and Shparlinski and Rakhmonov. The new ingredients include a reworking of Burgess' elementary methods for estimating mean values of character sums and a new estimate for bilinear sums of character sums which makes use of duality for certain complete character sums.