L^q Norms of Fekete and Related Polynomials

Abstract

Abstract. 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 L q norm on the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and ( · | p) is the Legendre symbol (so that z −1 fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of L q and L 2 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 L 4 norm of these polynomials.

Moments of polynomials with random multiplicative coefficients

“…where ( ⋅ 𝑝 ) is the Legendre symbol. Recently, building on work in [3,18], Günther and Schmidt [13] established the 𝑝 → ∞ limiting value of moments…”

“…For primes $$p$$, they are the $$p-1$$ degree polynomial defined by $$\begin{equation*} F_p(\theta ) = \frac{1}{\sqrt {p-1}}\sum _{n=1}^{p-1} {\left(\frac{n}{p}\right)} e(n\theta ), \end{equation*}$$where $$(\frac{\cdot }{p})$$ is the Legendre symbol. Recently, building on work in [3, 18], Günther and Schmidt [13] established the $$p\rightarrow \infty$$ limiting value of moments $$\begin{equation*} \int _0^1 {\left| F_p(\theta ) \right|}^{2k}\,d\theta , \end{equation*}$$for all fixed integers $$k$$. The moments are not Gaussian.…”

Improved Lower Bound for the Mahler Measure of the Fekete Polynomials

“…The L q norm of polynomials related to Fekete polynomials were studied in several recent papers. See [8,10,11,23,25,26], for example. An interesting extremal property of the Fekete polynomials is proved in [9].…”

Power series approximations to Fekete polynomials

“…Starting with the work of Conrey, Granville, Poonen and Soundararajan [7], much attention has been devoted to analytic properties of these polynomials, such as the distribution of zeros and relations between various norms, see [8,9,10,11] and references therein.…”