Pär Kurlberg: An introduction to the large sieve

The large sieve can be viewed as an analogue of Bessel's inequality for a set of non-orthormal vectors. Often it can profitably be applied in the following setting: to a sequence of "frequencies" in [0,1] we can associate a family of exponential sums; if the frequencies are "well separated", the large sieve gives that most exponential sums are fairly small. This innocent sounding conclusion has very powerful consequences, we can for instance deduce Bombieri's famous result that GRH, the generalized Riemann hypothesis, holds on average. (For many arithmetic applications, this is essentially as good as knowing GRH.)