A notorious problem in analytic number theory is to understand the distribution of squares modulo primes p. For example, if n₂(p) denotes the least positive quadratic nonresidue modulo p, then Vinogradov conjectured that n₂(p) is eventually smaller than p^ϵ, for any fixed ϵ > 0. We are far from proving this; the state-of-the art in this direction is a result of Burgess, now fifty years old, that any ϵ > (4)^-1 is permissible.

Instead of looking at the maximal order of n₂(p), one might consider the average size of n₂(p). This was determined by Erds in 1961: The mean value of n₂(p) is the finite constant ∑ _k=1^∞p_k∕2^k ≈ 3.675, where p_k denotes the kth prime in increasing order. In this talk, we discuss some recently obtained extensions and generalizations of Erdos’s result. In particular, we sketch a determination of the size of the the average least non-split prime in a cubic number field. Some of these results represent joint work with Greg Martin.