Merit Factors of Character Polynomials

Let q be a prime and be a non-principal character modulo q.Let where 1 t q is the character polynomial associated to (cyclicallypermuted t places). The principal result is that for any non-principaland non-real character modulo q and 1 t q, where the implicit constant is independent of t and q. Here||·||₄ denotes the L₄ norm on the unit circle. It follows from this that all cyclically permuted characterpolynomials associated with non-principal and non-real charactershave merit factors that approach 3. This complements and completesresults of Golay, Høholdt and Jensen, and Turyn (andothers). These results show that the merit factors of cyclicallypermuted character polynomials associated with non-principalreal characters vary asymptotically between 3/2 and 6. The averages of the L₄ norms are also computed. Let q be a primenumber. Then where the summation is over all characters modulo q.