2016/02/04 by Christian Günther, Günther, Christian, Kai‐Uwe Schmidt +1 · 1 citation
Mathematics · #11B83 #30C10 #42A05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1602.01750
openalex publication_date 2016/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in \-1,1\. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq norm on the complex unit circle. We consider the Fekete polynomials fp(z)=∑j=1p-1(j| p) zj, where p is an odd prime and ( ⋅| p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp(z) when q is an even positive integer and p→∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.