Merit Factors of Character Polynomials

Let q be a prime and χ be a non‐principal character modulo q . Letf x t ( z ) : = ∑ n = 0 q ‐ 1 χ ( n + t ) z nwhere 1 ⩽ t ⩽ q is the character polynomial associated to χ (cyclically permuted t places). The principal result is that for any non‐principal and non‐real character χ modulo q and 1 ⩽ t ⩽ q ,‖ f χ t ( z ) ‖4 4 = 4 3 q 2 + O ( q 3 / 2log 2 q )where the implicit constant is independent of t and q . Here ∥·∥ 4 denotes the L 4 norm on the unit circle. It follows from this that all cyclically permuted character polynomials associated with non‐principal and non‐real characters have merit factors that approach 3. This complements and completes results of Golay, Høholdt and Jensen, and Turyn (and others). These results show that the merit factors of cyclically permuted character polynomials associated with non‐principal real characters vary asymptotically between 3/2 and 6. The averages of the L 4 norms are also computed. Let q be a prime number. Then∑ χ ( mod q )‖ f χ t ‖ 4 4 = ( 2 q ‐ 3 )( q ‐ 1 ) 2where the summation is over all characters modulo q .