© 2000 by London Mathematical Society
© The London Mathematical Society
Merit Factors of Character Polynomials
Department of Mathematics and Statistics, Simon Fraser University Burnaby, BC, Canada V5A 1S6
Department of Mathematics, University of Hong Kong Pokfulam Road, Hong Kong
Received 11 November 1998. Revision received 26 January 1999.
Let q be a prime and 
be a non-principal character modulo q. Let
 |
where 1 
t q is the character polynomial associated to (cyclically permuted t places). The principal result is that for any non-principal and 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 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 L4 norms are also computed. Let q be a prime number. Then
 |
where the summation is over all characters modulo q.