*Caveat lector*: as an answer to Timothy's question, this is tangential at best. 

Regarding Gerry and Kevin's comments, people might be interested in Section 2 of [these notes of mine][1]
from an undergraduate number theory course, in which the philosophy of "almost square root error" is batted around for a while, especially with regard to the Riemann hypothesis.  This is maybe my favorite set of lecture notes from this course, perhaps because I got a chance to (talk and) write excitedly about things I hardly understand: I certainly do not claim a professional level of insight here.  (Indeed some regulars on this site could do far better, and I would be interested to hear their critiques.) 

The upshot though is that of agreement with Gerry and Kevin: having size $\sqrt{p}$ **on the nose** is just a little bit better than random, but the little bit makes a big difference: to me this suggests very precise structure rather than randomness.

Please [see here][2] for another take on the subject of almost square root error in the context of the Sato-Tate Conjecture.  This latter article was written at almost the same time as mine, and the author happens to be my former thesis advisor.  I am reasonably sure that both of these coincidences are indeed coincidental.








[1]: http://alpha.math.uga.edu/~pete/4400pnt.pdf

[2]: http://www.math.harvard.edu/~mazur/papers/nature_sato_tate.pdf