Notes on G2, determinants, and equidistribution

We determine the exact shape of the G2 equidistribution law for the one parameter family of exponential sums over , Here χ2(x) denotes the quadratic character (x/p), t in is the parameter, and p is any prime other than 2 or 7. This answers a question raised in Keating et al. (J. Phys. A Math. Gen. 36 (2003) 2943, footnote 3) and in Serre (pers. commun., March 7, 2002). We also analyze the analogous families when 7 is replaced by any odd integer n 3.