In joint work with Christopher Frei we use the circle method to estimate averages of arithmetic functions over values of random integer polynomials.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/14023