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MERIT FACTORS OF CHARACTER POLYNOMIALS

Published online by Cambridge University Press:  01 June 2000

PETER BORWEIN  and
KWOK-KWONG STEPHEN CHOI
PETER BORWEIN
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
KWOK-KWONG STEPHEN CHOI
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
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Abstract

Let q be a prime and χ be a non-principal character modulo q. Let

formula here

where 1 [les ] t [les ] 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 [les ] t [les ] q,

formula here

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

formula here

where the summation is over all characters modulo q.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 61 , Issue 3 , June 2000 , pp. 706 - 720
Copyright
The London Mathematical Society 2000

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