We determine the exact shape of the G2 equidistribution law for the one parameter family of exponential sums over Fp ×, ∑ x mod p, x≠0 χ2(x) exp(2πi(x7+tx)/p). Here χ2(x) denotes the quadratic character (x/p), t in Fp, 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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Algebra and Number Theory
- Applied Mathematics