In recent joint works with Stephan Baier, we were able to improve the large sieve inequalities for square moduli. The result is better, in certain ranges, than all previously known results which were obtained both jointly and independently by Baier and myself. I shall speak about the history, heuristics and conjectures about this problem and the techniques that enabled us to obtain the new result. I will also talk about application which is an approximation to the n^2+1 problem.

In that direction, I will also talk about a recent result of ours regarding primes in quadratic progressions on average.